\section{Introduction}

The growth of ultra-thin Ag films on Pt(111) has received much attention in the past few decades~\cite{Davies1982, Paffett1985, Roder1993a, Hartel1993, Grossmann1996, Schuster1996, Becker1993}, caused mainly by complex surface alloying~\cite{Roder1993, Struber1993, Bendounan2012} resulting in the formation of stress stabilized surface nanostructures~\cite{Zeppenfeld1994, Zeppenfeld1995, Tersoff1995, Jankowski2014}. The interest was further raised by the possibility to generate and tune novel chemical and physical properties by varying stoichiometry at the surface and careful control of the Ag growth conditions~\cite{Jankowski2014, Diemant2015, Schuettler2015}, and the formation of periodic dislocation networks~\cite{Brune1994, Ait-Mansour2012, Jankowski2014a, Hlawacek2015} used as nano-templates~\cite{Brune1998} for the growth of organic films~\cite{Ait-Mansour2006, Ait-Mansour2008, Ait-Mansour2009, Ait-Mansour2009a}.

Low-energy electron diffraction (LEED)~\cite{Paffett1985}, thermal energy atom scattering (TEAS)~\cite{Becker1993} and scanning tunnelling microscopy (STM)~\cite{Roder1993a} experiments revealed that at room temperature the first Ag layers grew through the formation of large pseudomorhic and thus strained Ag islands. This strain is caused by an about 4\% lattice mismatch between the lattice constants of bulk Ag and Pt. An increase of the surface temperature above 550~K leads to irreversible disorder at the surface~\cite{Becker1993}. STM investigations~\cite{Roder1993} revealed that this disorder is caused by the formation of a surface confined alloy~\cite{Tersoff1995} comprised of strained nanometre-sized Ag-rich structures embedded in the Pt surface~\cite{Roder1993}. The shape and size of these structures varied strongly with coverage~\cite{Zeppenfeld1994, Zeppenfeld1995, Jankowski2014}. After deposition of one monolayer~(ML) of Ag the surface was found to dealloy and a pseudomorphic Ag layer was found. Further deposition of Ag induced the formation of a triangular dislocation network which allows relief of the surface strain~\cite{Brune1994}. The third layer was reported to be a pure silver layer~\cite{Ait-Mansour2012} with a propagation of a height undulation originating from the dislocation network at the buried interface~\cite{Ait-Mansour2012, Jankowski2014a, Hlawacek2015}.

Much important microscopic work on the growth of ultra-thin silver films on Pt(111) has been done since this system has become a prototype of surface confined alloying.  However, in-situ spatio-temporal information during deposition is still completely lacking. The present study fills this remarkable gap. As will be shown the system shows incredibly rich dynamic behaviour which provides important details of the (de\=/)alloying processes in this system. Our investigation reveals that during growth of the first layer at 750-800~K a AgPt surface alloy forms with areas exhibiting a different AgPt-stochiometry. Beyond about 0.5~ML added Ag, the surface starts to dealloy which becomes prominent near coalescence of the first layer islands. Further increase of the Ag coverage results in the formation of irregularly shaped vacancy islands which are gradually filled by Pt atoms expelled from the alloy phase. In the last stage, these segregated Pt atoms form compact clusters which are observed as black spots in low energy electron microscopy (LEEM) images. These areas have also been analysed \textit{ex~situ} with atomic force microscopy (AFM). The combination of these techniques leads to our identification of these black spots as heavily strained and possibly even amorphous Pt features. When the coverage reaches 1~ML we observe that the integral area of these clusters decreases, which is attributed to their partial dissolution caused by a reentrant alloying of the ultrathin Ag-film. These ``amorphous'' Pt clusters are visible on the surface up to a coverage of several layers and are stable upon exposure to atmospheric conditions.

\section{Experimental}

The LEEM measurements were performed with an Elmitec LEEM~III with a base pressure better than $1 \times 10^{-10}$~mbar. The used Pt(111) crystal had a miscut angle of less than 0.1$^{\circ}$~\cite{Linke1985}. Surface cleaning was done by prolonged repetitive cycles of argon ion bombardment, annealing in oxygen at $2 \times 10^{-7}$~mbar at 800~K, and subsequent flashing to 1300~K in the absence of oxygen. The sample was heated by electron bombardment from the rear and the temperature was measured with a W3\%/Re-W25\%/Re thermocouple. An Omicron EFM-3 evaporator was used to deposit 99.995\% purity silver from a molybdenum crucible onto the sample. The coverage was calibrated on the basis of recorded LEEM movies, where we could track growth of the first pseudomorphic silver layer followed by the growth of second layer which propagates from step edges. One monolayer (ML) is defined as a layer with an atomic  density equal to that of a Pt(111) layer. LEEM images were recorded in bright-field mode using an appropriate contrast aperture around the (00) diffracted beam in the focal plane. The low energy electron diffraction (LEED) patterns were recorded using the largest available aperture of 25~$\rm\mu$m in the illumination column.

The AFM measurements were done under ambient conditions with an Agilent~5100 AFM employing amplitude-modulation for recording height topography. A MikroMasch Al-black-coated NSC35 Si$_{3}$N$_{4}$ AFM tip with a tip radius of 8~nm was used in these measurements. The resonance frequency of this tip was 205~kHz and the nominal spring constant 8.9~N/m. For the measurements an amplitude set-point of 90\% was used and an oscillation amplitude in the range from 30 to 40~nm. The amplitude modulation imaging mode provides  simultaneously topographic and phase images. The latter provides information on the local variation in energy dissipation involved in the contact between the tip and the sample. Various factors are known to influence this energy dissipation, among which are viscoelasticity, adhesion and chemical composition~\cite{Garcia2007}.

\section{Results and Discussion}
\subsection{An overview of initial growth and alloying of Ag on Pt(111); LEEM}

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.6]{leem_frames_alloying.eps}
\caption{LEEM images recorded in bright-field mode at coverage:(a)~0.275~ML, (b)~0.38~ML, (c)~0.5~ML, (d)~0.76~ML, (e)~1 ML, (f)~1.25~ML. The red arrows in (c) points emerging ``black'' spots. The FoV is 2~$\rm{\mu}$m, the electron energy is 2~eV and the substrate temperature is 750~K. The small dark spots, best visible in fig.~(a) are LEEM channel plate defects.}
\label{fig:leem_frames_alloying}
\end{figure*}

Figure~\ref{fig:leem_frames_alloying} shows a sequence of LEEM images recorded during the growth of the first Ag layer on Pt(111) at 750~K. The brightness of the various features in these successively recorded images cannot be compared due to the digital enhancement of each individual image in order to obtain always the best contrast settings. An image with a 2~$\rm\mu$m field of view (FoV) of the initial Ag deposition on Pt(111) surface is shown in Fig.~\ref{fig:leem_frames_alloying}(a). The results have been obtained with a deposition rate of $1.7\times10^{-3}$~ML/s. Figure~\ref{fig:leem_frames_alloying}(a) shows a large terrace in the centre, bordered by ascending monoatomic steps. Upon deposition the growth of a surface confined alloy is observed, starting from the ascending step edges~\cite{Roder1993a} and followed by the nucleation and growth of alloyed islands seen in Fig.~\ref{fig:leem_frames_alloying}(a)\=/(b). Beyond a Ag coverage of roughly 0.5~ML [Fig.~\ref{fig:leem_frames_alloying}(c)] the surface dealloys~\cite{Zeppenfeld1994} slowly and the islands start to coalesce. During the dealloying phase the surface is quite heterogeneous, especially but not exclusively, on top of the largest islands. Darkish or ``black'' spots [marked by red arrows in Fig.~\ref{fig:leem_frames_alloying}(c)] develop first in the centre of the adatom islands and later at an enhanced rate during coalescence. The black dots reveal Pt segregation upon which we focus on in this paper, not only on the process but also on its physical background ascription. At 0.75~ML, coalescence of the islands leads to formation of elongated vacancy clusters [meandering ``lines'' in Fig.~\ref{fig:leem_frames_alloying}(d)]. With continuing Ag deposition these vacancy clusters are filled by expelled Pt atoms and at 1 ML [Fig.~\ref{fig:leem_frames_alloying}(e)] they then take compact shapes as we will detail further below. Completion of the monolayer leads to a decrease of the integral exposed area of the black dots. This shrinkage of the dots is accompanied with reduced total brightness and indicates re-entrant alloying. The second layer starts to grow by step-propagation [see Fig.~\ref{fig:leem_frames_alloying}(f)] and island nucleation in the center of terraces.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{black_spots_area.eps}
\caption{Integrated fractional area of the black spots~vs~coverage. The fractional area is calculated from the images recorded with 4~$\rm{\mu}$m FoV.}
\label{fig:black_spots_area}
\end{figure*}

To illustrate the evolution of the Pt segregation we plot in Fig.~\ref{fig:black_spots_area} the fractional area of the black spots as a function of the coverage. The initial increase of the area, up to 0.8~ML, is caused by development of the black spots in the centres of the islands. Later, the steep increase, at around 0.85~ML, is related to the filling of the vacancies by Pt atoms at an enhanced rate, which is followed by a distinct decrease just before completion of the first layer. This decrease reveals the partial dissolution of the black spots due to the re-entrant alloying of the film when approaching completion of the first monolayer.

\subsection{Mixing in initial sub-monolayer growth of Ag on Pt(111) ; LEEM}

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{esther_initial.eps}
\caption{LEEM images recorded in bright-field mode :(a)~clean~Pt(111), (b)~0.3~ML, (c)~0.48~ML, (d)~0.7~ML. The FOV is 2~$\rm{\mu}$m, the electron energy is 2.0~eV and the substrate temperature is 750~K. The small dark spots, best visible in fig.~(a) are LEEM channel plate defects.}
\label{fig:esther_initial}
\end{figure*}

Figure~\ref{fig:esther_initial} shows representative snapshots of a movie taken during the initial growth of Ag/Pt(111). The brightness of the exposed layer decreases immediately after opening of the Ag evaporator shutter. This is indicative of an increase of the diffuse scattering due to alloying related disorder~\cite{Becker1993} and will be discussed in more detail in the next section. It is also obvious that at first the material is accommodated at the ascending steps. These steps propagate downward toward the large terrace in the center. Initially effectively all material, i.e., the excess Ag atoms and the Pt atoms expelled from the exposed surface are sufficiently mobile to reach the pre-existing steps. During these initial stages the brightness of the surface is identical everywhere indicating that the emerging surface undergoes alloying in which the Ag embedded in the surrounding Pt matrix~\cite{Roder1993,Zeppenfeld1994,Zeppenfeld1995} is homogeneously distributed. There is also no brightness variation on the narrower terraces near the outer skirts of the images. After some incubation time of about 180~s (0.3~ML) a few islands, visible in Fig.~\ref{fig:esther_initial}(b), nucleate in the central region of the large lower terrace. This provides another indication of a peculiar, alloying related feature. Where initially the mobility was sufficiently large for the atoms to reach the bordering steps, the actual mobility becomes insufficient to reach the propagating steps even if their separation has decreased. This is the first indication for a decreasing mobility of (Pt- and Ag) ad-atoms on the alloying surface. Further evidence for this process is obtained upon a comparison of Figs.~\ref{fig:esther_initial}(c)-(d).

\begin{figure*}[h!]
\centering
\includegraphics[scale=1.3]{esther_graph_islands.eps}
\caption{The island size distribution for various submonolayer coverages of 0.30, 0.45 and 0.60 monolayers presented in (a), (b) and (c), respectively}
\label{fig:esther_graph_islands}
\end{figure*}

The number of islands increases up to very late stages of growth as illustrated in Fig.~\ref{fig:esther_graph_islands}. Unlike in conventional nucleation and growth where nucleation is about finished after deposition of 1\% of a monolayer and the emerging islands keep growing with a quite narrow size distribution, until they coalesce~\cite{Venables2000}. In the present case nucleation still is active at~≈~0.60~ML. Moreover, the island size distribution is extremely broad and positional distribution is all but homogeneous. The smallest islands emerge near the atomic steps in the originally denuded zone.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{esther_graph_N_islands.eps}
\caption{Number of first layer islands as a function of deposition time. The deposition rate is $1.7\times10^{-3}$~ML/s.}
\label{fig:esther_graph_N_islands}
\end{figure*}

Another way of illustrating the late nucleation behaviour is depicted in Fig.~\ref{fig:esther_graph_N_islands}, where we plotted the number of islands as a function of time. Progressive nucleation takes place, which leads to a peak in the island density at 64\% of a monolayer. A straightforward explanation for these combined observations is alloying in the two exposed levels. Due to the progressive heterogeneity of the top layer the effective diffusion length decreases continuously and dramatically. As a result, enhanced nucleation takes place. In a site hopping framework one would conclude that the activation energy for diffusion increases by not less than a factor of 4. One safely arrives at the conclusion that initially the mobility of the ad-atoms decreases with increasing coverage. However, an attempt to nail this down to more definite numbers will fail due to the complexity of the system. The diffusing species are both Ag atoms and expelled Pt atoms and these move across a heterogeneous top-layer with small Ag-rich patches embedded in the Pt matrix. Moreover, with increasing coverage the concentration of embedded Ag atoms increases, while in a site hopping model the residence times on top of Ag filled sites and on top of Pt filled sites will most likely be different for diffusing Ag atoms and Pt atoms. One cannot exclude either that exchange processes further slow down the diffusion rates. We note that the de-mixing occurring at higher coverage leads to a more homogeneous composition and, as a consequence of reversing the argument, to enhanced diffusion and thus exclusion of new nucleation events. During these late stages of monolayer growth the number of islands also decreases as a result of coalescence processes.

\subsection{De-alloying of the alloy above 0.5 ML ; LEEM}

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{esther_spots_higher_ML.eps}
\caption{LEEM images recorded in bright-field mode :(a)~1.0~ML, (b)~1.68~ML. The FOV is 2~$\rm{\mu}$m, the electron energy is 2.0~eV and the substrate temperature is 750~K.}
\label{fig:esther_spots_higher_ML}
\end{figure*}

A striking feature is the emergence of black spots as shown in Fig.~\ref{fig:esther_spots_higher_ML}(a)-(b). The black spots do appear when approaching completion of the monolayer and are emerging in Fig.~\ref{fig:esther_initial}(c). They persist after the growth of several monolayers, and are even visible after the completion of the eighth monolayer (not shown here). We have carefully tried to identify their origin. For this purpose we have varied the deposition rate between $1.7\times10^{-4}$ and $1.1\times10^{-2}$~ML/s. Both the number density of black dots and their integrated area are at variance with the anticipated behaviour for contamination: both increase a few tens of a percent, i.e., much less than the factor 13, which might be anticipated on the exposed time of the, compared to Ag, by far more reactive Pt\=/surface layer. Occasionally, we do observe some evidence of carbon impurities (not visible in presented LEEM figures). However, their signature as a result of creating under-focus and over-focus conditions differs completely from that of the black dots in Fig.~\ref{fig:esther_initial}~and~Fig.~\ref{fig:esther_spots_higher_ML}. We therefore conclude that the ``black spots'' are inherent to the Ag-Pt(111) de-mixing process. They appear to consist of disordered Pt-rich patches which reduce in integral size with increasing film thickness, but persist even after deposition of 8 ML.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{esther_area_plot.eps}
\caption{(a) the growth process of Fig.~\ref{fig:esther_initial} analysed in the uncovered area (see text). (b) the mean bright field intensity as a function of time. The exposed toplayer of the substrate is represented by the squares, the exposed first layer by circles. The area taken by black spots is subtracted from the analysis. The break above 400 s is caused by not-well defined morphology of the surface caused by rapid de-alloying which causes big uncertainties in the perfomed analysis.  Imaging electron energy is 2 eV.}
\label{fig:esther_area_plot}
\end{figure*}

Figure~\ref{fig:esther_area_plot}(a) shows the evolution of what naively would be called the uncovered area, i.e., the supposedly still exposed virgin Pt(111) surface. As we will see shortly this conjecture cannot be maintained. The area taken by the black dots is subtracted. Within the uncertainties originating from this subtraction and/or from lensing effects as a result of field irregularities due to e.g. work function differences, the ``exposed Pt'' area decreases linearly with time, indicative of the growth of a single layer. In this particular experiment it takes about 590 s to complete the first layer. More information is obtained from the averaged brightness of the layers evolving at the two exposed levels as shown in Fig.~\ref{fig:esther_area_plot}(b). In this evaluation the black dots have been disregarded. The grey symbols refer to the original level of the clean Pt(111) surface, while the black circles represent the results for the exposed surface at a coverage that is one monolayer higher. As becomes immediately evident the brightness decays strongly from the very start of the deposition. It passes through a pronounced minimum and subsequently increases again up to a maximum value after roughly 500~s~(0.85~ML) (as we will explain below this maximum brightness marks busy activity in de-mixing, segregation, stress relaxation and re-entrant mixing). It is also obvious that the average brightness of both exposed layers is very similar. Both observations are completely consistent with the formation of a mixed ad-layer, or the surface confined alloy discussed in Ref.~\cite{Roder1993,Zeppenfeld1995,Zeppenfeld1994}. Alloying proceeds during the deposition of the first half layer, while de-alloying occurs during deposition of the second half monolayer. The alloy can only persist in the vacuum exposed layer(s) due to the involved energetics. First the tensile tension in the Pt(111) is reduced by the incorporation of the larger Ag atoms~\cite{Tersoff1995}, while with increasing Ag content the tension finally becomes compressive, which leads to the exchange of first layer Pt with Ag embedded in the lower layer. In the initial phase the brightness decays due to increasing disorder by the continuing embedding of Ag patches in the Pt\=/matrix, while later on the de-mixing leads to lesser disorder: the continuing expulsion of Ag leads to larger Pt patches surrounding the embedded Ag.

\begin{figure*}[h!]
\centering
\includegraphics[scale=1.5]{esther_rims_plot.eps}
\caption{The width of the bright border around the large islands has
been measured of a number of islands of several deposition experiments.
There is a strongly proffered.}
\label{fig:esther_rims_plot}
\end{figure*}

The brightness across the level of the growing ad-layer is by no means homogeneously distributed during the growth beyond about 0.40 ML. A representative snapshot is shown in Fig.~\ref{fig:esther_initial}(c). At the chosen imaging conditions the smaller islands are bright, the ones with ``intermediate'' sizes are slightly less bright, many of them have a distinct black dot, and some of them have a distinctly brighter rim, when compared to their centres. This feature is even clearer for the largest islands, which all have a bright rim and a dimmer central part. We interpret these findings as follows: the brighter areas represent Ag(-rich) areas. The smaller islands which appear during late stages of the nucleation are constituted by Ag atoms which are still continuously deposited and Ag atoms which are now expelled from the initial level, i.e., from the alloyed exposed parts of the substrate. During later stages of growth of the intermediately sized and largest islands, the edges also accommodate mostly Ag-atoms for the same reason. The resulting bright rims appear to have a characteristic width. The results obtained for a number of islands from several deposition experiments is shown in Fig.~\ref{fig:esther_rims_plot}. The distribution of the widths of these bright rims/borders is somewhat skewed but has a pronounced maximum at about 52~nm. The fact that these widths show no distinct correlation with island size or deposition rates suggests their thermodynamic origin. We attribute this to the relaxation of stress. The compressed Ag-film can best (partly) relieve its stress near the descending step edges. The emerging of bright rims is responsible for the slightly higher brightness obtained for the highest layer around 300~-~400~s in Fig.~\ref{fig:esther_area_plot}(b).

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{esther_last_stage.eps}
\caption{LEEM images recorded in bright-field mode :(a)~0.75~ML, (b)~0.82~ML, (c)~0.88~ML, (d)~0.95~ML. The FOV is 2~$\rm{\mu}$m, the electron energy is 2.0~eV and the substrate temperature is 750~K.}
\label{fig:esther_last_stage}
\end{figure*}

With increasing silver deposition the de-mixing of the surface confined Ag/Pt alloy continues. The expulsion of surplus Ag from the lower level, i.e., the original substrate level, is apparently facilitated by the advancing steps. However, the Pt atoms, encapsulated on the intermediate and in particular on large islands lack that opportunity and experience difficulties to escape. Therefore, these centres remain relatively dark during a prolonged period. Upon proceeding de-mixing also the larger islands become brighter, the segregation intensifies and dark grey spots evolve, signalling segregation. On the largest islands, the dimmer inner parts are also Pt-rich and strong fluctuations occur as evidenced in Fig.~\ref{fig:esther_last_stage}. The dark grey areas fluctuate strongly both in position and also in brightness. Finally also the largest islands host dark grey spots [cf. Figure~\ref{fig:esther_initial}~(d) taken at a coverage of 0.7 ML]. We note that at the same time the brightness of the small islands is unaffected. Some of the dark grey areas evolve into dark spots, while others seem to dissolve again in the Ag-layer. Around a total Ag coverage of 0.85 ML, i.e., during late stages of coalescence an enormous bustle is taking place. The irregularly shaped vacancy clusters, which are the result of the continuing growth of the coalesced islands, are with overwhelming majority filled by segregation of Pt into black dots [see Fig.~\ref{fig:esther_last_stage}~(c)-(d) and Fig.~\ref{fig:fig_03_bs}]. At the same time the brightness of the large islands has now reached its maximum [see Fig.~\ref{fig:esther_area_plot}~(b)]. This late and strikingly active stage of the birth of new Pt-rich dots happens precisely in those areas where the step advancement ceases. This filling of the irregularly shaped vacancy islands with black Pt-dots goes along with a strong reduction of their integrated length, since all dots are circular in shape. The energetically unfavourable long integrated step length apparently facilitates the Pt segregation after which a much reduced domain border length surrounding the circular black spots, has developed. Both features are quantified and illustrated in Fig.~\ref{fig:fig_03_bs} and Fig.~\ref{fig:fig_04_bs} .

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{fig_03.eps}
\caption{LEEM images recorded in bright-field mode at coverage: (a)~0.75~ML, (b)~0.79~ML, (c)~0.82~ML, (d)~0.85~ML, (e)~0.89~ML, (f)~0.92~ML , (g)~0.96~ML,  (h)~1~ML. The electron energy is 1.9~eV }
\label{fig:fig_03_bs}
\end{figure*}

We stress that we did carefully look for indications of structures in the recorded $\mu$LEED patterns during Ag deposition, different from the pseudomorphic (1x1) structure and did not observe any evidence for those. At this same time, as we will show in a moment, we observe relaxations of the surface lattice which reaches maximum at 0.85 ML (see Fig.~\ref{fig:leed_spots}), the coverage where the brightness of the islands reaches maximum in Fig.~\ref{fig:esther_area_plot}~(b). We therefore must conclude that these black dots are probably amorphous Pt(-rich?) areas. Further support for this conclusion is provided by their circular shape: The Pt-rich dots are heavily strained in their Ag-rich environment and undergo strong and isostropic, compressive stress. In a later stage some even coalesce and become larger, eventually resuming their circular shape. We observe that the decrease of the size of the vacancies goes along with a change of their shape to a more compact one. This shape evolution can be seen in Fig.~\ref{fig:fig_04_bs} which shows the compactness of the vacancies as a function of coverage. The compactness $C$ is defined as:

\begin{equation*}
C = \frac{(A\cdot4\pi)^\frac{1}{2}}{P}
\end{equation*}
where $P$ is the perimeter of vacancies and $A$ is their area.

At 0.85~ML presented in Fig.~\ref{fig:fig_03_bs}(d), the expelled Pt atoms start to segregate into the vacancies, which finally at 1~ML [see~Fig.~\ref{fig:fig_03_bs}(h)] leads to their complete filing. We note two more things: First with increasing compactness the length of the energetically less favourable edges decreases. Also the step energy will be lower in contact with platinum compared to a vacancy. Both features lead to a lowering of the total energy and thus facilitate the Pt segregation. In other words under heavy compressive stress the exposed Pt atoms actually experience an energetic advantage by segregation into the compacting vacancies. We note that, as illustrated in Fig.~\ref{fig:fig_04_bs}, the compactness never seems to reach its ideal value~of~1. We attribute this to difficulties inherent to setting a well defined discrimination level. Uncertainties related to field distortion effects~\cite{Nepijko2001} play a role and a principal problem is caused by the digital nature of the image. With a finite pixel size one cannot define a line with infinite sharpness. As a result the determined $C$ will always be an underestimate of the real compactness.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{fig_04.eps}
\caption{Plot of vacancies compactness as a function of coverage.}
\label{fig:fig_04_bs}
\end{figure*}

\subsection{LEED spots intensity variation and surface lattice relaxation}

\begin{figure*}[h!]
\centering
\includegraphics[scale=1]{leed_intensity.eps}
\caption{Intensity of  LEED spots as a function of coverage: (a)-(00) spot, (b)-(10) spot and (c)-(01) spot. Height of the LEED spots as a function of coverage: (e)-(00) spot, (f)-(10) spot and (g)-(01) spot. Blue line marks completion of the first monolayer. Fig.~(d) represents ratio of the (10) and (01) spots intensity as a function of coverage. Fig.~(h) presents typical LEED pattern recorded from clean Pt(111) with labelled position of corresponding spots. Electron energy is 44~eV, sample temperature is 800~K.}
\label{fig:leed_intensity}
\end{figure*}

The surface lattice variation can be studied in detail from the evolution of the \textit{in~situ}  $\mu$LEED patterns recorded as function of Ag coverage. In the initial phase of Ag deposition, surface alloying induces the sharp decrease of the intensity of the Bragg spots~\cite{Becker1993,Jankowski2014a} shown in Fig.~\ref{fig:leed_intensity}, but their position corresponding to the Pt(111) lattice [Fig.~\ref{fig:leed_spots}] is not altered. At around 0.7 ML, where coalescence of islands and nucleation of well defined black spots in their centres is observed, the intensity of the (01) spots decreases to reach local minimum at 0.85 ML. At this same coverage, the (01) spot intensity reaches a maximum and both first order spots broaden slightly towards the (00) spot, as shown in Fig.~\ref{fig:leed_spots}(c). We attribute this broadening to the relaxation of the surface lattice towards the 4\% larger Ag distances. This relaxation of the lattice is triggered by de-alloying of the surface and partial segregation of Pt into vacancy clusters as discussed above. Moreover, the value of (10)/(01) spots intensity ratio plotted as a function of coverage in Fig.~\ref{fig:leed_spots}(d) reaches minimum at exactly 0.85 ML. This is a direct result of a transient lowering of the threefold symmetry: The Ag-atoms in the relaxed film not only occupy FCC sites but also HCP sites. The latter contribute to enhanced sixfold symmetry features.

The first monolayer is pseudomorphic, as its observed p(1$\times$1) LEED pattern given in Fig.~\ref{fig:leed_spots}(e) corresponds to that of the Pt(111) lattice \cite{Paffett1985,Becker1993,Jankowski2014a}. We also mention that after reaching 0.85 ML the intensity of the (00) spot in Fig.~\ref{fig:leed_spots}(a) again decreases, which rationalized as re-entrant mixing of the top layer. This is in line with the partial dissolution of the black spot [cf. Fig.~\ref{fig:black_spots_area}] and the concomitant decrease of the brightness of exposed layers [Fig.~\ref{fig:esther_area_plot}~(b)]

\begin{figure*}[h!]
\centering
\includegraphics[scale=1]{leed_spots.eps}
\caption{The (01) LEED spot recorded during growth of Ag on Pt(111) at 800 K: (a) clean Pt(111), (b) at 0.65 ML, (c) at 0.74 ML, (d) at 0.85 ML, (e) at 1 ML. The arrow in figure (a) points the direction towards the (00) spot. The energy of electrons was 44 eV. The intensity of the spots is presented on logarithmic scale of colours.}
\label{fig:leed_spots}
\end{figure*}

We now focus on the nature of the black spots and apply different techniques for their characterization. The latter include spatially resolved work function (WF) measurements with LEEM and \textit{ex situ} AFM data. The latter provides support for the conclusion that excreted Pt atoms constitute the black dots.
\subsection{Work function changes}

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.9]{WF_images.eps}
\caption{LEEM images recorded in bright-field mode at the indicated deposition time of silver (grey scale colour) and the concomitant change in the WF relative to clean Pt(111) (colour scale images). The deposition time needed for completion of the first layer is around $\sim$30~min}
\label{fig:WF_images}
\end{figure*}

The LEEM offers a capability of recording spatial maps of relative changes of the surface WF change (see also Ref.~\cite{Hlawacek2015} for more details) during deposition of thin films. Figure~\ref{fig:WF_images} shows a sequence of LEEM images and the corresponding spatial maps of the surface WF change as a function of deposition time.  The atomically clean terraces of Pt(111) exhibit a constant value of the WF and only a 0.2~eV lower WF is seen at the step edges \cite{Besocke1977}. The initial deposition of Ag leads to a sharp decrease of the average WF, shown in Fig.~\ref{fig:fig_07_bs}. After 10~min of Ag-deposition (at around $\sim$0.3~ML) the WF of the growing islands ($\rm{\Delta\phi=-0.79}$~eV) is 0.1~eV lower than WF of the surrounding hosting terrace ($\rm{\Delta\phi=-0.69}$~eV). This contrast in WF is attributed to different levels of intermixing of Ag and Pt in the islands and their surroundings. The work function has been measured also above the black spots, assumed to be Pt(-rich) clusters. As shown in Fig.~\ref{fig:fig_07_bs}, the latter follows quite closely the WF variation of their environment at a distance of about -0.2~eV. The similarity in temporal dependence makes sense if one realizes that the Pt(-rich) clusters are small and long range interactions play a major role in determining the work function. This seems somewhat surprising at first sight since the WF of clean and smooth Pt (111) (5.8~eV) is much larger than that of clean and smooth Ag(111) (4.56~eV)~\cite{Kawano2008}. However, the WF depends strongly on the surface orientation with a tendency for lower WF observation with decreasing coordination, i.e, $\rm{\Delta\phi_{111} < \Delta\phi_{100} <\Delta\phi_{110}}$. This is even worse for amorphous dots with which we most likely deal with. Finite size effects may also lower the WF~\cite{Nepijko2001}. It is noted, that field distortions around small objects make drawing clear conclusions hazardous. Therefore, we tentatively conclude that the case for Pt as the constituting material of the clusters remains undecided on the basis of the current WF data.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.75]{fig_07.eps}
\caption{Average and Pt clusters work function as a function of coverage.}
\label{fig:fig_07_bs}
\end{figure*}

A further increase of the coverage leads to a decrease of the average WF which at coverage of 1 and 2~ML has the value of $\rm{\Delta\phi=-1.085}$~eV and $\rm{\Delta\phi=-1.33}$~eV, respectively. The rather constant value of the WF beyond two layers thick films indicates that at most minor changes in topography and composition of the surface occur beyond a film thickness of two layers.

At a coverage of 3~ML the average WF saturates at $\rm{\Delta\phi=-1.37}$~eV. This saturation value is in the range reported by Hartel~et~al.~\cite{Hartel1993}~(-1.2~eV) and by Paffet~et~al.~\cite{Paffett1985}~(\=/1.5~eV) obtained at similar coverages. We note that the reported values were measured for the alloy formed after annealing the surface at 600~K. Schuster~et~al.~\cite{Schuster1996} showed, that both temperature and time during alloying drastically influence intermixing levels of the Ag/Pt(111) surface alloy, which results in different surface morphology leading to a variation in the surface WF. Moreover, the step density of the initial Pt(111) surface may have a decisive influence of several tenths of an eV on the measured value \cite{Poelsema1982}. In that respect, we can safely conclude that our data compare favourably with the macroscopic data available in literature.

\subsection{Characterization of 3~ML thick film in ambient AFM}

To gain more insight on the composition and (if possible) structure of the black spots we apply AFM. For that purpose we prepared a 3~layers thick Ag/Pt(111) film by deposition in UHV at 800~K as monitored by LEEM. As evident from Fig.~\ref{fig:WF_images}, and the discussion above, the surface predominantly consists of silver, but a substantial number of black spots is still observed. The AFM data have been taken in ambient atmosphere.

\begin{figure*}[h!]
\centering
\includegraphics[scale=0.6]{fig_08.eps}
\caption{AFM images of 3~ML Ag deposited on Pt(111) at 800~K and subsequently cooled down to room temperature. The measurements were done at ambient conditions. (a)~topography image , (b)~phase contrast image, (c)~typical line profiles trough clusters obtained from phase contrast (red line) and topography (black line) image; the profiles have been shifted vertically for sake of clarity .}
\label{fig:fig_08_bs}
\end{figure*}

Figure~\ref{fig:fig_08_bs} shows AFM data of the surface of the 3~ML Ag/Pt(111) system grown in UHV (see above) and measured under ambient conditions. The topography image of the surface presented in Fig.~\ref{fig:fig_08_bs}(a) shows step contrast which indicates that the surface is still flat. No pronounced geometric structures are observed. The phase contrast image presented in Fig.~\ref{fig:fig_08_bs}(b), reveals the presence of many small, $\sim$50~nm wide regions for which the measured phase shift is 10$^{\circ}$ lower than for the rest of the surface. We attribute these regions to the Pt clusters observed with LEEM as the number density of the clusters obtained from the AFM- and the LEEM-images is very similar (8-15 clusters/$\rm{\mu}$m$^{2}$). From the typical height profiles shown in Fig.~\ref{fig:fig_08_bs}(c) measured along the lines drawn in Fig.~\ref{fig:fig_08_bs}(a)~and~(b) we concluded that Pt clusters are about one to two layers higher than the surrounding layer. The phase shift contrast in the recorded images can be attributed to many factors~\cite{Garcia2007}, nevertheless it points to that the chemical composition of the clusters differs from the majority at the surface, which at 3~ML is mainly composed of Ag atoms. We stress that the correlation between the phase contrast and the height contrast is perfect. The black protrusions have a different chemical composition and under the applied ambient conditions probably oxidized Pt is the natural candidate. Therefore, the conclusion that the black dots consist of mainly platinum atoms is straightforward. The fact that the clusters protrude from the surface also explains the difficulty of selecting a proper focus condition in LEEM for these objects. We do believe that the footing of these clusters is still at the substrate and an oxygen layer contributes to the height differences.

A careful look at Fig.~\ref{fig:fig_08_bs}(b) shows a pretty smooth edge on the upper terrace side of the ascending step edge. This is probably related to the initial step propagation growth mode~\cite{Jankowski2014b, Vroonhoven2005}. The area has apparently a slightly different composition and or structure. We ascribe this feature to different possibilities of dealing with stresses near a descending step.

\section{Conclusions}

The spatio-temporal \textit{in situ} information from LEEM on the growth of ultrathin Ag on Pt(111) at 750-800~K shows vivid and rich dynamical behaviour. We confirm alloying and subsequent de-alloying during the growth of the first monolayer at these elevated temperatures. Initially alloying occurs and nucleation processes are prolonged as a result of increasingly reduced mobility of Ag and Pt ad-species on the alloying surface. New nucleation events are even observed at a coverage exceeding 70\% of a monolayer! The ad-islands have a wide size distribution and are quite heterogeneous, especially the larger ones. Beyond a coverage of 50\% of a monolayer a violent segregation of Pt towards the centre of the islands occurs and bright Ag-rims become apparent. De-alloying is fast during coalescence of the adatom islands. The remaining irregularly shaped vacancy islands are quickly filled by Pt(-rich) patches (black dots) and take a compact circular shape. These features are energetically favoured since the integral step length is decreased and the boundaries between Pt(-rich) and Ag(-rich) areas are minimized. A concomitant transient relaxation of the exposed layer is deferred from the diffraction profiles. Upon further completion of the monolayer alloying re-enters and the initial pseudo-morphological structure is resumed as accompanied by re-entrant (partial) alloying, which continues in the second layer.

The black dots are attributed to stressed, possibly amorphous Pt(-rich) features. \textit{Ex situ} AFM experiments support this picture as the observed phase contrast images suggest chemical contrast. These Pt-features persist even after deposition of several monolayers and probably have a firm base at the Pt-substrate.

\section{Acknowledgments}

We want to thank Robin Berkelaar for acquiring the AFM data. This work is part of ECHO research program 700.58.026, which is financed by the Chemical Sciences Division of the Netherlands Organisation for Scientific Research(NWO).

\clearpage
\bibliographystyle{iopart-num}
{\footnotesize


\section{Introduction}
\label{sec:1}
A smooth flow $(\phi_t)$ generated by a smooth vector field $X$ on a
compact manifold~$M$ is called \emph{stable} if the range of the Lie
derivative $\mathcal L_X: C^\infty(M) \to C^\infty(M)$ is closed and
it is called \emph{cohomology-free} or \emph{rigid} if it is stable
and the range of the Lie derivative operator has codimension one.  The
\emph{Katok (or Katok-Hurder) conjecture} \cite{K01}, \cite{K03},
\cite{H85} states that every cohomology-free vector field is smoothly
conjugate to a linear flow on a torus with Diophantine frequencies. It
is not hard to prove that all cohomology-free flows are volume
preserving and uniquely ergodic (see for instance \cite{Forni}).

We also recall that the Katok conjecture is equivalent to the
\emph{Greenfield-Wallach conjecture}~\cite{GW} that every globally
hypoelliptic vector field is smoothly conjugate to a Diophantine
linear flow (see \cite{Forni}). A smooth vector field $X$ is
called~\emph{globally hypoelliptic} if any $0$-dimensional current $U$
on $M$ is smooth under the condition that the current $\mathcal L_X U$
is smooth. Greenfield and Wallach in~\cite{GW} proved this conjecture
for homogeneous flows on \emph{compact} Lie groups.  (The equivalence
of the Katok and Greenfield-Wallach conjectures was essentially proved
already in \cite{CC00} as noted by the third author of this paper.
The details of the proof can be found in~\cite{Forni}).

The best general result to date in the direction of a proof is the
joint paper of the third author \cite{HH} where it is proved that a
every cohomology-free vector field has a factor smoothly conjugate to
Diophantine linear flow on a torus of the dimension equal to the first
Betti number of the manifold $M$. This result has been developed
independently by several authors \cite{Forni}, \cite{Kocsard},
\cite{Matsumoto} to give a complete proof of the conjecture in
dimension $3$ and by the first author in the joint paper
\cite{FlaminioPaternain} to prove that every cohomology-free flow can
be embedded continuously as a linear flow in a possibly non-separated
Abelian group.

From the definition, it is clear that there are two main mechanisms
which may prevent a smooth flow from being cohomology-free: it can
happen that the flow is not stable or it can happen that the closure
of its range has codimension higher than one. For instance, linear
flows on tori with \emph{Liouvillean} frequencies are not stable with
range dense in a closed subspace of dimension one (the subspace of
functions of zero average), while translation flows \cite{Forni97},
\cite{MMY}, horocycle flows \cite{FF1} and nilflows \cite{FF} are in
general stable but have range of countable codimension.

Until recently, basically no other examples were known of non
cohomology-free smooth flows. In particular there was no example of
the first kind of phenomenon, that is, of flows which are not stable
with range closure of codimension one, except for flows smoothly
conjugate to \emph{linear} Liouvillean toral examples. In the past
couple of years several examples of this kind have been constructed by
Avila and collaborators.  Avila and Kocsard \cite{AvilaKocsard} have
proved that every non-singular smooth flow on the $2$-torus with
irrational rotation number has range closed in the subspace of
functions of zero average with respect to the unique invariant
probability measure.  Avila and Kocsard and Avila and Fayad have also
announced similar examples on certain higher dimensional compact
manifolds, not diffeomorphic to tori, which admit a non-singular
smooth circle action (hence Conjecture 6.1 of \cite{Forni} does not
hold). In all these examples the closure of the range of the Lie
derivative operator on the space smooth functions has codimension one,
or equivalently, the space of all invariant distributions for the flow
has dimension one (hence it is generated by the unique invariant
probability measure). We recall that an \emph{invariant distribution}
for a flow is a distribution (in the sense of S.~Sobolev and
L.~Schwartz) such that its Lie derivative along the flow vanishes in
the sense of distributions.

The goal of this paper is to prove that examples of this kind do not
exist among \emph{homogeneous flows}, so that a non-toral homogeneous
flow always fails to be cohomology-free also because the closure of
its range has codimension higher then one. In fact, we prove that for
any homogeneous flow on a \emph{finite-volume} homogeneous manifold
$M$, except for the case of flows smoothly isomorphic to linear toral
flows, the closure of the range of the Lie derivative operator on the
space of smooth functions has countable codimension, or, in other
terms, the space of invariant distributions for the flow has countable
dimension. As a corollary we have a proof of the Katok and
Greenfield-Wallach conjectures for general homogeneous flows on
finite-volume homogeneous manifolds.  Our main result can be stated as
follows.

\begin{theorem}
  \label{thm:Main}
  Let $G/D$ a connected finite volume homogeneous space. A homogeneous
  flow $(G/D, g^\field{R})$ is either smoothly isomorphic to a linear flow on
  a torus or it has countably many independent invariant
  distributions.
\end{theorem}

An important feature of our argument is that in the case of
\emph{partially hyperbolic} flows we prove the stronger and more
general result that any partially hyperbolic flow on any compact
manifold, not necessarily homogeneous, has infinitely many distinct
minimal sets (see Theorem~\ref{expandingminimal}). In particular, we
have a proof of the Katok and Greenfield-Wallach conjectures in this
case.  We are not able to generalize this result to the finite-volume
case. However, we can still prove that a partially hyperbolic
\emph{homogeneous} flow on a finite-volume manifold has countably many
ergodic probability measures (see Proposition~\ref{prop:5:2}).

In the non partially hyperbolic homogeneous case, that is, in the
quasi-unipotent case, by the Levi decomposition we are able to reduce
the problem to flows on semi-simple and solvable manifolds.  The
semi-simple case is reduced to the case of $\mathrm S \mathrm L_2(\field{R})$ by an
application of the Jacobson--Morozov's Lemma which states that any
nilpotent element of a semi-simple Lie algebra can be embedded in an
$\sl_2(\field{R})$-triple. The solvable case can be reduced to the nilpotent
case for which our main result was already proved by the first two
authors in~\cite{FF}.  In both these cases the construction of
invariant distributions is based on the theory of unitary
representations for the relevant Lie group (Bargmann's classification
for $\mathrm S \mathrm L(2,\field{R})$ and Kirillov's theory for nilpotent Lie groups).

The paper is organized as follows. In section~\ref{sec:2} we deal with
partially hyperbolic flows on compact manifolds.  In
section~\ref{sec:3} give the background on homogeneous flows that
allows us to reduce the analysis to the solvable and semi-simple
cases.  A further reduction is to consider quasi-unipotent flows
(sect.~\ref{sec:4}) and partially hyperbolic flows (sect.~\ref{sec:5}) on finite-volume non-compact manifolds; then the main
theorem follows easily (sect.~\ref{sec:6}).  Finally, in section~\ref{sec:7} we state
a general conjecture on the stability of homogeneous flows and a couple of more general related open problems.

\smallskip
{\bf Acknowledgments} Livio Flaminio was supported in part by
the Labex~CEMPI (ANR-11-LABX-07). Giovanni Forni was supported by NSF
grant DMS~1201534. Federico Rodriguez Hertz was supported by NSF
grant DMS~1201326. Livio Flaminio would also like to thank the Department
Mathematics of the University of Maryland, College Park, for its hospitality
during the preparation of this paper.

\section{Partially hyperbolic flows on compact manifolds}
\label{sec:2}

The goal of this section is to prove the following theorem.

\begin{theorem}
  \label{expandingminimal}
  Let $M$ be a compact connected manifold, $\phi^t$, $t\in\field{R}$, a flow
  on $M$ and assume $\phi^t$ leaves invariant a foliation $\mathcal F$
  with smooth leaves and continuous tangent bundle, e.g.\ the unstable
  foliation of a partially hyperbolic flow. Assume also that the flow
  $\phi^t$ expands the norm of the vectors tangent to $\mathcal F$
  uniformly. Then there are infinitely many different $\phi^t$-minimal
  sets.
\end{theorem}

The existence of at least one non trivial (i.e.\ different from the
whole manifold) minimal sets goes back G.~Margulis (see
\cite{Starkov}, \cite{MR96k:22022}, \cite{Kleinbock-Shah-Starkov})
and Dani \cite{MR794799},
\cite{MR870710}. A similar idea was already used by R.~Ma\~n\'e in
\cite{MR516217} and more recently by A.~Starkov \cite{Starkov}, and
F. \& J.~Rodriguez Hertz and R.~Ures \cite[Lemma A.4.2 (Keep-away
Lemma)]{MR2390288}, in different contexts.

\smallskip Theorem~\ref{expandingminimal} will follow almost
immediately from the next lemma.

\begin{lemma}
  \label{induct}
  Let $\phi^t$ be a flow like in Theorem~\ref{expandingminimal}. For
  any $k$-tuple $p_1,\dots, p_k\in M$ of points in different orbits
  and for any open set $W\subset M$, there exist $\epsilon>0$ and
  $q\in W$ such that $d(\phi^t(q),p_i)\geq\epsilon$ for all $t\geq 0$
  and for all $i=1,\dots,k $.
\end{lemma}

Let us show how Theorem~\ref{expandingminimal} follows from
Lemma~\ref{induct}.

\begin{proof}[Proof of Theorem~\ref{expandingminimal}]
  Since $M$ is compact there is a minimal set $K$. Assume now by
  induction that there are $K_1,\dots K_k$ different minimal set, then
  we will show that there is a minimal set $K_{k+1}$ disjoint from the
  previous ones. Let $p_i\in K_i$, $i=1,\dots, k$ be $k$ points and
  take $q$ and $\epsilon>0$ from Lemma~\ref{induct}. Since
  $d(\phi^t(q),p_i)\geq\epsilon$ for any $t\geq 0$ and for $i=1,\dots,
  k$ we have that for any $i=1,\dots, k$, $p_i\notin\omega(q)$, the
  omega-limit set of $q$. Since the $K_i$'s are minimal, this implies
  that $K_i\cap \omega(q)=\emptyset$ for $i=1,\dots, k$. Take now a
  minimal subset of $\omega(q)$ and call it $K_{k+1}$.
\end{proof}

Let $F=T\mathcal F$ be the tangent bundle to the foliation with fiber
at $x\in M$ given by $F(x)=T_x\mathcal F(x)$. We denote by $d$ the
distance on $M$ induced by some Riemannian metric. Let $X$ be the
generator of the flow $\phi^t$.  Let also
$E(x)=(F(x)\oplus\<X(x)\>)^{\perp}$ be the orthogonal bundle and
$\mathcal E_r(x)=\exp_x(B^E_{r}(x))$ be the image of the $r$ ball in
$E(x)$ by the exponential map.  Let $f$ be the dimension of the
foliation $\mathcal F$ and $m$ the dimension of $M$. For $r \le r_0$
the disjoint union $\mathcal E:=\sqcup_{x\in M} \mathcal E_r(x)$ is a
$(m-f-1)$-dimensional continuous disc bundle over~$M$. Denote with
$d_{\mathcal F}$ and $d_{\mathcal E}$ the distances along the leaves
of $\mathcal F$ and $\mathcal E$, and let
\[
\mathcal F_r(x)=\{ y\in \mathcal F(x) \mid d_{\mathcal F}(y,x) \leq
r\} \subset \mathcal F(x)
\]
be the $f$-dimensional closed disc centerd at $x$ and of radius $r>0$.
Clearly $d\le d_{\mathcal F}$ and $d\le d_{\mathcal E}$.

We may assume that the Riemannian metric on $M$ is adapted so that
$\mathcal F_r(x) \subset\phi^{-t} \mathcal F_r(\phi^t x)$ for all
$x\in M$ and $r,t\ge 0$. In fact if $g$ is a Riemannian metric such
that $\| \phi^{t}_* v \|_g \ge C \lambda^t \|v\|_g$ for all $v \in
F(x)$, all $x\in M$ and all $t\ge 0$, (where $\lambda >1$), then
setting $\hat g = \int^{T_0}_0 (\phi^t)^* g \, \hbox{d} t$, with $T_0=
- \log_\lambda (C/2)$, we have that, for all $v \in F(x)$ and $x\in
M$, the function $\| \phi^{t}_* v\|_{\hat g}$ is strictly increasing
with $t$.

We may choose $r_1< r_0$ such that if $r \le r_1$ then, for all $x\in
M$, we have $d_{\mathcal F}(y,z) \le 2 d(y,z)$ for any $y,z\in
\mathcal F_r(x)$ and $d_{\mathcal E}(y,z) \le 2 d(y,z)$ for any
$y,z\in \mathcal E_r(x)$.

For $x\in M$, let
\[
V_{\delta,r}(x)=\bigcup_{z\in \mathcal E_{\delta}(x)}\mathcal F_r(z).
\]
There exists $ r_2 \le r_1$ be such that for all $r,\delta\le r_2$
then $V_{\delta, 4 r}(x)$ is homeomorphic to a disc of dimension
$(m-1)$ transverse to the flow.

\smallskip\noindent{\bf Normalization assumption}: After a constant
rescaling of $X$ we may assume that given any $x\in M$,
$z,y\in\mathcal F(x)$ and $t\geq 1$ we have $d_{\mathcal
  F}(\phi^t(z),\phi^t(y))\geq 4\,d_{\mathcal F}(z,y)$. Henceforth, in
this section, we shall tacitly make this assumption.

\begin{proof}[Proof of Lemma~\ref{induct}]
  Let $p_1,\dots, p_k\in M$ be points belonging to different orbits
  and let $W\subset M$ be an open set.  We shall find $r>0$ and a
  point $x_0 \in W$ with $ \mathcal F_r(x_0)\subset W$ and then
  construct, by induction, a sequence of points $x_n\in M$ and of
  iterates $\tau_n\geq 1$ satisfying, for some $\delta >0$, the
  following conditions
  \begin{equation}
    \label{eq:1}\tag{$A_{n+1}$}
    \phi^{-\tau_n}(\mathcal F_r(x_{n+1}))\subset \mathcal
    F_r(x_n), \qquad  \text{\ for all\ } n\ge 0,
  \end{equation}
  and
  \begin{equation}
    \label{eq:2}\tag{$B_n$}
    \phi^{T_n}x\in \mathcal
    F_r(x_n)\implies \phi^t(x)\notin\hbox{$\bigcup_i$}V_{\delta,
      2r}(p_i)\,, \quad \text{\ for all\ } t
    \in [0, T_{n+1}).
  \end{equation}
  Here we have set $T_n:=\sum_{k=0}^{n-1}\tau_k$. Then defining $D_n:=
  \phi^{-T_n}\mathcal F_r(x_n)$ we have $D_{n+1}\subset D_n\subset
  \mathcal F_r(x_0)$ and any point $q\in\bigcap_n D_n\subset \mathcal
  F_r(x_0)\subset W$ will satisfy the statement of the Lemma.

  By the choice of an adapted metric we have
  $$
  d_{\mathcal F} (\phi^t (x),\phi^t(p)) \ge d_{\mathcal F}(x,p) \,,
  \qquad \text{ for all } p\in M \text{ and all } x\in \mathcal F(p)
  \,.
  $$
  This implies that for all $i=1, \dots ,k$ any $r>0$ and any $t\ge 0$
  \begin{equation}
    \label{eq:leaf_incl}
    x\in \mathcal F_{4r}(p_i)\setminus\mathcal F_{2r}(p_i) \implies
    \phi^t(x)\not \in \mathcal F_{2r}(\phi^t(p_i)).
  \end{equation}
  Hence there exists $\delta_0 < r_2$ such that for all $\delta <
  \delta_0$ and all $r\le r_2$ we have:
  \begin{enumerate}[(i)]
  \item\label{item1} for all $i\in \{1,\dots ,k\}$,
    \[
    \phi^{[0,1]}\big( V_{\delta, 4r}(p_i) \setminus V_{\delta,
      2r}(p_i)\big) \cap V_{\delta, r}(p_i)= \emptyset \,.  \]
  \end{enumerate}
  The above assertion follows immediately by continuity if $p_i$ is
  not periodic of minimal period less or equal to $1$. If $p_i$ is
  periodic of period less or equal to $1$, then it follows by
  continuity from formula~\eqref{eq:leaf_incl}.

  As the orbits of $p_1, \dots, p_k$ are all distinct and the set $W$
  is open, we may choose a point $x_0\in W$ and positive real numbers
  $r,\delta< \delta_0$ so that the following conditions are also
  satisfied:
  \begin{enumerate}[(i)]
    \setcounter{enumi}{1}
  \item\label{item2} $\mathcal F_r(x_0)\subset W$;
  \item\label{item3} for all $i, j \in \{1,\dots ,k\}$, with $i\not=
    j$,
  $$
  \phi^{[0,1]} \big( V_{\delta, 4r}(p_i)\big) \cap \phi^{[0,1]} \big(
  V_{\delta, 4r}(p_j)\big)= \phi^{[0,1]} \big( V_{\delta,
    4r}(p_i)\big) \cap \bigcup_{t\in [0,1]} \mathcal F_r(\phi^tx_0)
  =\emptyset\,.
  $$
\end{enumerate}

If for all $t>0$ we have $ \mathcal F_r(\phi^t(x_0))\cap
\hbox{$\bigcup_i$}V_{\delta, r}(p_i) =\emptyset$, then $d(\phi^t(x_0),
p_i)> r$ for all $i=1,\dots k$ and all $t>0$, proving the Lemma with
$q=x_0$ and $\epsilon =r$. Thus we may assume that
  $$
  \tau_0:=\inf\Big\{t>0\;:\;\mathcal F_r(\phi^t(x_0))\cap
  \hbox{$\bigcup_i$}V_{\delta, r}(p_i) \neq\emptyset\Big\} < \infty
$$
and define
$$  \hat x_0 = \phi^{\tau_0}(x_0).
$$
The above condition~\eqref{item3} implies that $\tau_0\ge 1$, hence by
the normalisation assumption it follows that
\begin{equation}
  \label{eq:3}
  \mathcal F_{5r}(\hat x_0) \subset  \phi^{\tau_0}\big(\mathcal F_r(x_0)\big).
\end{equation}

Assume, by induction, that points $x_k\in M$ and iterates $\tau_k\geq
1$ satisfying the conditions~({$A_n$}) and~\eqref{eq:2} have been
constructed for all $k\in \{0,\dots, n\}$, and assume that the point
$\hat x_n:= \phi^{\tau_n}(x_n)\in M$ is such that $\mathcal F_r(\hat
x_n)$ intersects non-trivially some disc $ V_{\delta,r}(p_i) $.  Since
$V_{\delta, r}(p_i)$ is saturated by $\mathcal F$, it follows that
$\mathcal F_{2r}(\hat x_n)\cap \mathcal E_{\delta}(p_i)$ consists of a
unique point $z_n$ with $d_{\mathcal F}(z_n, \hat x_n) \le 2 r$; we
define $x_{n+1}\in \mathcal F(\hat x_n)$ as the point at distance $3r$
on the geodesic ray in $\mathcal F(\hat x_n)$ going from $z_n$ to $
\hat x_n$ (or any point on the geodesic ray issued from $z_n$ if $\hat
x_n=z_n$). Then we have
\begin{equation}
  \label{eq:4}
  \mathcal F_ r(x_{n+1})\subset \mathcal F_{4r}(
  \hat x_n )\cap V_{\delta,4r}(p_i)\setminus
  V_{\delta,2r}(p_i)\,.
\end{equation}
Since $\mathcal F_ r(x_{n+1})\subset V_{\delta, 4r}(p_i)$, for all
$t\in (0,1)$ we have
  $$
  \mathcal F_r(\phi^t x_{n+1}) \subset \phi^t \mathcal F_r( x_{n+1})
  \subset \phi^{[0,1]}\big( V_{\delta,4r}(p_i) \setminus V_{\delta,
    2r}(p_i)\big)\,.
  $$
  By the disjointness conditions~\eqref{item1} and~\eqref{item3}, it
  follows that, for all $t\in (0, 1]$,
  $$\mathcal F_r(\phi^t x_{n+1})\cap \bigcup_{i=1}^k V_{\delta,
    r}(p_i) =\emptyset\,.$$

  It follows that if we define
  \[
  \tau_{n+1}:=\inf \Big\{t>0\;:\;\mathcal F_r(\phi^t(x_{n+1}))\cap
  \hbox{$\bigcup_iV_{\delta, r}(p_i)$}\neq\emptyset\Big\},\qquad \hat
  x_{n+1}= \phi^{\tau_{n+1}}(x_{n+1})
  \]
  (assuming $\tau_{n+1}<+\infty$), then $\tau_{n+1}\ge 1$, and by the
  normalisation assumption and by the inclusion in
  formula~\eqref{eq:4} we have
  \[
  \mathcal F_{r}(x_{n+1}) \subset \mathcal F_{4r}(\hat x_{n})=
  \mathcal F_{4r}(\phi^{\tau_{n}} x_{n}) \subset
  \phi^{\tau_{n}}\big(\mathcal F_r(x_{n})\big)
  \]
  and by construction, having set $T_{n+2}:=\sum_{k=0}^{n+1}\tau_k$,
  we also have
 $$
 x\in D_{n+1}:=\phi^{-T_{n+1}}(\mathcal F_r(x_{n+1}))\implies
 \phi^t(x)\notin\hbox{$\bigcup_i$}V_{\delta, 2r}(p_i)\,, \quad \text{
   for all } t \in [0, T_{n+2}).
$$
The inductive construction is thus completed.  As we explained above
we have that~$(D_n)$ is a decreasing sequence of closed sub-intervals of
$\mathcal F_r(x_0)$ and that any point $q\in \bigcap_n D_n$ satisfies
$\phi^t(q) \notin \bigcup_i V_{\delta, r}(p_i)$ for all $t\ge 0$.

The above inductive construction may fail if at some stage $n\geq 0 $
we have $\tau_n = +\infty$. In this case let $q$ be any point in
$\phi^{-T_n}(\mathcal F_r(x_n))$. Again such a point $q\in W$
satisfies the statement of the Lemma, hence the proof is completed.
\end{proof}

\section{Homogeneous flows}
\label{sec:3}

Henceforth $G$ will be a connected Lie group and $G/D$ a finite volume
space; this means that $D$ is a closed subgroup of $G$ and that $G/D$
has a finite $G$-invariant (smooth) measure. The group $D$ is called
the \emph{isotropy group} of the space $G/D$.

With $g^\field{R}$ we denote a one-parameter subgroup $(g^t)_{t\in \field{R}}$ of
$G$. The flow generated by this one-parameter subgroup on the finite
volume space $G/D$ will be denoted $(G/D, g^\field{R})$ or simply $g^\field{R}$.

\begin{remark}
  \label{rem:3:1}
  Our first observation is that, in proving Theorem~\ref{thm:Main}, we
  may suppose that the flow~$g^\field{R}$ is ergodic on $G/D$ with respect to
  a finite $G$-invariant measure. This is due to the fact that the
  ergodic components of the flow~$g^\field{R}$ are closed subsets of~$G/D$
  (see~\cite[Thm.\ 2.5]{Starkov}). Since $G/D$ is connected, either we
  have infinitely many ergodic components, in which case
  Theorem~\ref{thm:Main} follows, or the flow~$g^\field{R}$ is ergodic.
\end{remark}

Henceforth we shall assume that the flow~$g^\field{R}$ is ergodic. Whenever
convenient we may also assume that $G$ is simply connected by pulling
back the isotropy group $D$ to the universal cover of $G$.

Let $G=L\ltimes R$ be the Levi decomposition of a simply connected Lie
group $G$ and let $G_\infty$ be the smallest connected normal subgroup
of $G$ containing the Levi factor~$L$. Let $q: G\to L$ be the
projection onto the Levi factor. We shall use the following result.

\begin{theorem}[{\cite{MR0444835}, \cite{MR896893}, \cite[Lemma 9.4,
    Thm.\ 9.5]{Starkov}}]
  \label{thm:3:2}
  If $G$ is a simply connected Lie group and the flow $(G/D, g^\field{R})$ on
  the finite volume space $G/D$ is ergodic then
  \begin{itemize}
  \item The groups $R/R\cap D$ and $q(D)$ are closed in $G$ and in $L$
    respectively. Thus $G/D$ factors onto $L/q(D)\approx G/RD$ with
    fiber $R/R\cap D$.  The semi-simple flow $(L/q(D), q(g^\field{R}))$ is
    ergodic.
  \item The solvable flow $(G/\overline{G_\infty D}, g^\field{R})$ is
    ergodic.
  \end{itemize}
\end{theorem}

By Theorem~\ref{thm:3:2} it is possible to reduce the analysis of the
general case to that of the semi-simple and solvable cases. In fact,
the following basic result holds.

\begin{lemma}\label{lem:3:3}
  If the flow $(G/D, g^\field{R})$ smoothly projects onto a flow $(G_1/D_1,
  g_1^\field{R})$, via an epimorphism $p: G\to G_1$ with $D \subset
  p^{-1}(D_1)$, then the existence of countable many independent
  invariant distributions for the flow $(G_1/D_1, g_1^\field{R})$ implies the
  existence of countable many independent invariant distributions for
  the flow $(G/D, g^\field{R})$.
\end{lemma}
\begin{proof} We are going to outline two different proofs.

  \noindent {\it First Proof.\/} Let $\mu$ denote the invariant smooth
  measure on $G/D$, let $\mu_1$ the projected measure on
  $G_1/D_1$. For all $y\in G_1/D_1$, let $\mu_y$ denote the
  conditional measure on the fiber $p^{-1} \{y\} \subset G/D$ of the
  projection $p:G/D\to G_1/D_1$, which can be constructed as
  follows. Let $\omega$ be a volume form associated to $\mu$ on $G/D$
  and let $\omega_1$ be a volume form associated to $\mu_1$ on
  $G_1/D_1$. There exists a smooth form $\nu$ on $G/D$ (of degree $d$
  equal to the difference in dimensions between $G/D$ and $G_1/D_1$)
  such that $\omega = \nu \wedge p^\ast \omega_1$. Note that the form
  $\nu$ is not unique.  (In fact, it is unique up to the addition of a
  $d$-form $\eta$ such that $ \eta \wedge p^\ast
  \omega_1=0$). However, the restriction of the form $\nu$ to the
  fiber $p^{-1}\{y\}$ is a uniquely determined volume form on
  $p^{-1}\{y\}$ which induces the conditional measure $\mu_y$, for all
  $y\in G_1/D_1$.

  Since the measures $\mu$ is $G$-invariant and the measure $\mu_1$ is
  $G_1$-invariant, the family $\{ \mu_y \vert y\in G_1/D_1\}$ is
  smooth and $G_1$-equivariant. For any smooth function $f\in
  C^\infty(G/D)$ let $p_\# (f)\in C^\infty(G_1/D_1)$ be the smooth
  function defined as
 $$
 p_\# (f) (y) = \frac{1}{ \mu_y(p^{-1}\{y\})} \int_{ p^{-1}\{y\} }
 f(y) d\mu_y(x) \,, \quad \text{ for all } y\in G_1/D_1\,.
 $$
 The map $p_\# : C^\infty(G/D) \to C^\infty(G_1/D_1)$ is bounded
 linear and surjective.  In fact, it is a left inverse of the
 pull-back map $p^\ast: C^\infty(G_1/D_1) \to C^\infty(G/D)$.  For any
 $g_1^\field{R}$-invariant distribution $\mathcal D\in \mathcal D'(G_1/D_1)$,
 the following formula defines a $g^\field{R}$-invariant distribution
 $p^\#\mathcal D \in \mathcal D'(G/D)$:
 $$
 p^\# \mathcal D (f) = \mathcal D \left( p_\ast (f) \right) \,, \quad
 \text{ for all } f \in C^\infty(G/D)\,.
 $$
 The map $p^\#: \mathcal D'(G_1/D_1) \to \mathcal D'(G/D)$ is bounded
 linear and injective, since the map $p_\#$ is bounded linear and
 surjective, and by construction it maps the subspace of
 $g_1^\field{R}$-invariant distributions into the subspace of
 $g^\field{R}$-invariant distributions.

 \noindent {\it Second Proof.\/}  For any $G_1$-irreducible component
 $H$ of the space $L^2(G_1/D_1)$, the pull-back $p^\ast(H)$ is a $G$-
 irreducible component of the space $L^2(G/D)$. Since $p$ is an
 epimorphism, the space $C^\infty(p^\ast (H)) \subset p^\ast(H)$ of
 $G$-smooth vectors in $p^\ast(H)$ is equal to the pull-back $p^\ast
 C^\infty(H)$ of the subspace $C^\infty(H)\subset H$ of $G_1$-smooth
 vectors in $H$. Hence the push-forward map $p_\ast: C^\infty(p^\ast
 (H)) \to C^\infty(H)$ is well defined.  Assume that there exists a
 $g_1^\field{R}$-invariant distribution which does not vanish on
 $C^\infty(H)\subset C^\infty(G_1/D_1)$, then there exists a
 $g_1^\field{R}$-invariant distribution $\mathcal D_H$ which does not vanish
 on $C^\infty(H)$ but vanishes on $C^\infty(H^\perp)$ (which can be
 defined by projection). Let then $p^\ast (\mathcal D_H)$ the
 distribution defined as follows:
$$
p^\ast (\mathcal D_H)(f) = \begin{cases} \mathcal D_H (p_\ast f)\,,
  \quad &\text{\ if\ } f\in C^\infty(p^\ast (H)) \,;\\ 0\,, \quad
  &\text{\ if\ } f\in C^\infty(p^\ast (H)^\perp) \,.
\end{cases}
$$
By construction, the distribution $p^\ast (\mathcal D_H)$ is
$g^\field{R}$-invariant. Thus the one-parameter subgroup $g^\field{R}$ on $G/D$ has
infinitely many independent invariant distributions whenever the
one-parameter subgroup $g_1^\field{R}$ on $G_1/D_1$ does.
\end{proof}

In dealing with solvable groups it is useful to recall the theorem by
Mostow (see~\cite[Theorem E.3]{Starkov})

\begin{theorem}[Mostow]
  \label{thm:mostow} If $G$ is a solvable Lie group, then $G/D$ is of
  finite volume if and only if $G/D$ is compact.
\end{theorem}

When $G$ is semi-simple, in proving Theorem~\ref{thm:Main}, we may
suppose that $G$ has finite center and that the isotropy group $D$ is
a lattice. This is the consequence of the following proposition.

\begin{proposition}
  \label{prop:lattice_red}
  Let $G$ be a connected semi-simple group and let $G/D$ be a finite
  volume space.  If there exists an ergodic flow on $G/D$, then the
  connected component of the identity of $D$ in $G$ is normal in
  $G$. Hence we may assume that $G$ has finite center and that $D$ is
  discrete.
\end{proposition}

\begin{proof}
  We have a decomposition $G=K\cdot S$ of $G$ as the almost-direct
  product of a compact semi-simple normal subgroup $K$ and of a
  totally non-compact normal semi-simple group $S$. Let $p: G\to
  K_1:=G/S$ be the projection of $G$ onto the semi-simple compact
  connected group $K_1$.  Let $\bar g^t$ the flow generated by $\bar
  X=p_* X$ on the connected, compact, Hausdorff space
  $Y:=K_1/\overline{p(D)}$. As $Y$ is a homogeneous space of a compact
  semi-simple group, the fundamental group of $Y$ is finite. The
  closure of the one-parameter group $(\exp{t\bar X})_{t\in \field{R}}$ in
  $K_1$ is a torus subgroup $T<K_1$; it follows that the closures of
  the orbits of $\bar g^t$ on $Y$ are the compact tori $Tk
  \,\overline{p(D)}$, ($k\in K_1$), homeomorphic to $T/T\cap k
  \,\overline{p(D)}\, k^{-1}$.

  Let $g^\field{R}$ be an ergodic flow on $G/D$, generated by $X\in \mathfrak
  g_0$.  Since $\bar g^t$ acts ergodically on $Y$, the action of $T$
  on $Y$ is transitive. In this case we have $Y=T/T\cap
  \overline{p(D)}$, and since the space $Y$ is a torus with finite
  fundamental group, it is reduced to a point. It follows that $T<
  \overline{p(D)}=K_1$. Thus $p(D)$ is dense in $K_1=G/S$ and $SD$ is
  dense in $G$.

  Let $\bar D^Z$ denote the Zariski closure of $\mathrm A\mathrm d(D)$ in $\mathrm A\mathrm d(G)$
  (see~Remark 1.6 in \cite{MR591617}).  By Borel Density Theorem
  (see~\cite[Thm.~4.1, Cor.~4.2]{MR591617} and \cite{MR2158954}) the
  hypothesis that $G/D$ is a finite volume space implies that $\bar
  D^Z$ contains all hyperbolic elements and unipotent elements
  in~$\mathrm A\mathrm d(G)$. As these elements generate $\mathrm A\mathrm d(S)$, we have $\mathrm A\mathrm d(S) <
  \bar D^Z $, and the density of $SD$ in $G$ implies $\mathrm A\mathrm d(G)= \bar D^Z
  $. Since the group of $\mathrm A\mathrm d(g)\in \mathrm A\mathrm d(G)$ such that $\mathrm A\mathrm d(g)
  (\operatorname{Lie}(D)) = \operatorname{Lie}(D)$ is a Zariski-closed
  subgroup of $\mathrm A\mathrm d(G)$ containing $\bar D^Z$, we obtain that the
  identity component $D^0$ of~$D$ is a normal subgroup of~$G$ and
  $G/D\approx (G/D^0)/(D/D^0)$.  We have thus proved that we can
  assume that $D$ is discrete. We can also assume that $G$ has finite
  center since $D$ is a lattice in~$G$ and therefore it meets the
  center of $G$ in a finite index subgroup of the center.  This
  concludes the proof.
\end{proof}

Our proof of Theorem~\ref{thm:Main} considers separately the cases of
quasi-unipotent and the partially hyperbolic flows.  We recall the
relevant definitions.  \smallskip

Let $X$ be the generator of the one-parameter subgroup $g^\field{R}$ and
let~$\mathfrak g^\mu$ denote the generalised eigenspaces of eigenvalue
$\mu$ of $\mathrm a\mathrm d(X)$ on $\mathfrak g = \mathfrak g_0\otimes \field{C}$. The Lie
algebra $\mathfrak g$ is the direct sum of the $\mathfrak g^{\mu}$ and
we have $[\mathfrak g^\mu, \mathfrak g^\nu] \subset \mathfrak
g^{\mu+\nu}$. Let
\[
\mathfrak p^0 =\sum_{\Re \mu =0} \mathfrak g^\mu, \quad \mathfrak p^+
=\sum_{\Re \mu >0} \mathfrak g^\mu,\quad \mathfrak p^- =\sum_{\Re \mu
  <0} \mathfrak g^\mu,
\]
\begin{definition}
  \label{def:qu_ph}
  A flow $g^\field{R}$ on $G/D$ is called \emph{quasi-unipotent} if
  $\mathfrak g = \mathfrak p^0$ and it is \emph{partially hyperbolic}
  otherwise.  Thus the flow subgroup $g^\field{R}$ is quasi-unipotent or
  partially hyperbolic according to whether the spectrum of the group
  $\mathrm A\mathrm d(g^t)$ acting on $\mathfrak g$ is contained in $U(1)$ or not.
\end{definition}

\section{The quasi-unipotent case}
\label{sec:4}

We now assume that flow $g_\field{R}$ on the finite volume space is
quasi-unipotent.

\subsection{The semi-simple quasi-unipotent case}
\label{sec:G-semisimple}

In this subsection we assume that the group $G$ is semi-simple (and
the one-parameter subgroup $g^\field{R}$ is quasi-unipotent).

\begin{definition}
  An $\sl_2(\field{R})$ triple $(a, n^+, n^-)$ in a Lie algebra $\mathfrak
  g_0$ is a triple satisfying the commutation relations
  \[ [a, n^\pm]= \pm n^\pm, \quad [n^+,n^-]=a.
  \]
\end{definition}

We recall the Jacobson--Morozov Lemma~\cite{MR559927}.

\begin{lemma}[Jacobson--Morozov Lemma]
  \label{lem:Jacobson-Morozov}
  Let $n^+$ be a nilpotent element in a semi-simple Lie algebra
  $\mathfrak g_0$. Assume that $n^+$ commutes with a semi-simple
  element $s\in \mathfrak g_0$. Then in $\mathfrak g_0$ we can find a
  semi-simple element $a$ and a nilpotent element $n^-$ such that $(a,
  n^+, n^-)$ is an $\sl_2(\field{R})$ triple commuting with $s$.
\end{lemma}

Given a unitary representation of $(\pi, H)$ of a Lie group on a
Hilbert space $H$, we denote by $H^\infty$ the subspace of
$C^\infty$-vectors of $H$ endowed with the $C^\infty$ topology, and by
$(H^{\infty})'$ its topological dual.

\begin{lemma}[\cite{FF1}]
  \label{lem:4:1}
  Let $U^t$ be a unipotent subgroup of\/ $\hbox{\rm PSL}_2(\field{R})$. For each
  non-trivial irreducible unitary representation $(\pi, H) $ of\/
  $\hbox{\rm PSL}_2(\field{R})$ there exists a distribution, i.e.\ an element of $D\in
  (H^{\infty})'$, such that $U^t D = D$.
\end{lemma}

\begin{proposition}
  \label{prop:4:2}
  Let $G$ be semi-simple and let $D$ be a lattice in $G$. Then any
  quasi-unipotent ergodic subgroup $g^\field{R}$ of $G$ admits infinitely
  many $g_\field{R}$-invariant distributions on $C^\infty(G/D)$.
\end{proposition}

\begin{proof}
  By Proposition~\ref{prop:lattice_red} we may assume that $G$ has
  finite center and that $D$ is a lattice in $G$.  By the Jordan
  decomposition we can write $g^t= c^t \times u^t$ where $c^t$ is
  semi-simple and $u^t$ is unipotent with $c^\field{R}$, $u^\field{R}$ commuting
  one-parameter subgroups of~$G$. Since $c^\field{R}$ is semi-simple and
  quasi-unipotent, its closure in $G$ is a torus $T$.

  By the Jacobson--Morozov lemma we find an $\sl_2(\field{R})$ triple $(a,
  n^+, n^-)$ commuting with $c^\field{R}$, hence with $T$. We let $(a^t, u^t,
  v^t)$ be the corresponding one-parameter groups commuting with $T$
  and let $S$ the subgroup generated by these flows.

  It is well known that the center $Z(S)$ of $S$ is finite and that,
  consequently, there exists a maximal compact subgroup
  of~$S$. Indeed, the adjoint representation $\mathrm A\mathrm d_G|S$ of $S$ on the
  Lie algebra of $G$, as a finite dimensional representation of $S$,
  factors through $SL(2, \field{R})$, a double cover of $S/Z(S)$. The kernel
  of $\mathrm A\mathrm d_G|S$ is contained in $Z(G)$, because $G$ is connected. Since
  $Z(S)$ is monogenic, we have that $Z(G)$ is a subgroup of index one
  or two of $Z(G)Z(S)$. It follows that $Z(S)$ is finite.

  Let $K< S$ denote a maximal compact subgroup of $S$.  Let $
  L^2(G/D;\field{C})$ be the space of complex valued $L^2$ functions on
  $G/D$. The subspaces $H_0$ and $H$ of $ L^2(G/D;\field{C})$ formed
  respectively by the $T_1=T \cdot K$ and $T_2=T \cdot Z(S)$ invariant
  functions are infinite dimensional Hilbert spaces since the orbits
  of the subgroups $T_1$ and $T_2$ in $G/D$ are compact and the space
  $G/D$ is not a finite union of these orbits. Furthermore $H_0\subset
  H$ and $H$ is also invariant under~$S$. We deduce that $H$
  decomposes as a direct integral or direct sum of irreducible unitary
  representations of $S$; these representations are trivial on the
  center $Z(S)$ and therefore define irreducible unitary
  representations of ${\mathrm P\mathrm S\mathrm L}_2(\field{R})$. Since each
  irreducible unitary representation of ${\mathrm P\mathrm S\mathrm
    L}_2(\field{R})$ contains at most one $K$-invariant vector and $H_0$ is
  infinite dimensional, we have that the cardinality of the set of
  unitary irreducible representations of ${\mathrm P\mathrm S\mathrm
    L}_2(\field{R})$ appearing in $H$ is not finite. By the previous Lemma
  each unitary irreducible representation $H_i$ of ${\mathrm P\mathrm
    S\mathrm L}_2(\field{R})$ contained in $H$ contains a distribution
  $D_i\in (H_i^\infty)'$ which is $u^t$ invariant; this distribution
  is also $g^t$ invariant since the action of $c^t$ on $H$ is
  trivial. Since the space $H^\infty$ coincides with the Fr\'echet
  space of $C^\infty$ functions on $G/D$ which are $T \cdot Z(S)$
  invariant, the proposition is proved.
\end{proof}

\subsection{The solvable quasi-unipotent case}
\label{sec:solv-eucl-case}

In this subsection we assume that the group $G=R$ is solvable and the
one-parameter subgroup $g^\field{R}$ is quasi-unipotent and ergodic on the
finite volume space $R/D$.

\smallskip We recall the following definition.
\begin{definition} A solvable group $R$ is called a \emph{class
    $(\mathrm I)$ group} if, for every $g\in R$, the spectrum of
  $\mathrm A\mathrm d(g)$ is contained in the unit circle~$U(1)=\{z \in \field{C} \mid
  |z|=1\}$.
\end{definition}

It will also be useful remark that if $R$ is solvable and $R/D$ is a
finite measure space, then we may assume that $R$ is simply connected
and that $D$ is a quasi-lattice (in the language of Auslander and
Mostow, the space $R/D$ is then a \emph{presentation}); in fact, if
$\tilde R$ is the universal covering group of $R$ and $\tilde D$ is
the pull-back of $D$ to $\tilde R$, then the connected component of
the identity $\tilde D_0$ of $\tilde D$ is simply connected
\cite[Them. 3.4]{gorbatsevich1994lie}; hence $R/D\approx \tilde R
/\tilde D \approx R'/ D' $, with $R'=\tilde R/\tilde D_0$ solvable,
connected and simply connected and with $D'=\tilde D/\tilde D_0$ a
quasi-lattice.

We also recall the construction, originally due to Malcev and
generalised by Auslander \emph{et al.}, of the \emph{semi-simple} or
\emph{Malcev splitting} of a simply connected connected solvable group
$R$ (see \cite{MR0199308}, \cite{MR0486307}, \cite{MR0486308},
\cite{0453.22006}).

A solvable Lie group $G$ is \emph{split} if $G=N_G\rtimes T$ where
$N_G$ is the nilradical of $G$ and $T$ is an Abelian group acting on
$G$ faithfully by semi-simple automorphisms.

A \emph{semi-simple} or \emph{Malcev splitting} of a connected simply
connected solvable Lie group $R$ is a split exact sequence
\[
0 \to R \overset{m}{\to}M(R) \leftrightarrows T \to 0
\]
embedding $R$ into a split connected solvable Lie group $M(R)=
N_{M(R)} \rtimes T$ such that $M(R)= N_{M(R)} \cdot m(R)$; here
$N_{M(R)}$ and $T$ are as before. The image $m(R)$ of~$R$ is normal
and closed in $M(R)$ and it will be identified with $R$.

The semi-simple splitting of a connected simply connected solvable Lie
group $R$ is unique up to an automorphism fixing $R$.

Let $\operatorname{Aut}(\mathfrak r)\approx \operatorname{Aut}(R)$ be
automorphism group of the Lie algebra $\mathfrak r$ of $R$. The
adjoint representation $\mathrm A\mathrm d$ maps the group $R$ to the solvable
subgroup $\mathrm A\mathrm d(R)<\operatorname{Aut}(\mathfrak r)$; since
$\operatorname{Aut}(\mathfrak r)$ is an algebraic group we may
consider the Zariski closure $\mathrm A\mathrm d(R)^*$ of~$\mathrm A\mathrm d(R)$. The group
$\mathrm A\mathrm d(R)^*$ is algebraic and solvable, since it's the algebraic closure
of the solvable group $\mathrm A\mathrm d(R)$.  It follows that $\mathrm A\mathrm d(R)^*$ has a
Levi-Chevalley decomposition $\mathrm A\mathrm d(R)^* = U^*\rtimes T^*$, with $T^*$
an Abelian group of semi-simple automorphisms of $\mathfrak r$ and
$U^*$ the maximal subgroup of unipotent elements of $\mathrm A\mathrm d(R)^*$.

Let $T$ be the image of $\mathrm A\mathrm d(R)$ into $T^*$ by the natural projection
\hbox{$\mathrm A\mathrm d(R)^*\to T^*$}.  Since $T$ is a group of automorphisms of
$R$, we may form the semi-direct product $M(R)= R\rtimes T$.  By
definition we have a split sequence
$$
0 \to R \to M(R)\leftrightarrows T \to 0.
$$
It can be proved that $M(R)$ is a split connected solvable group
$N_{M(R)} \rtimes T$ and it is the semi-simple splitting of $R$
(see~.\ loc.\ cit.). We remark that the splitting $M(R)=N_{M(R)}
\rtimes T$ yields a new projection map $\tau:M(R)\to T$ by writing for
any $g\in M(R)$, $g = n\tau (g)$ with $n\in N_{M(R)}$ and $ \tau
(g)\in T$. Composing $\tau$ with the inclusion $R\to M(R)$ we obtain a
surjective homomorphism $\pi:R\to T$.

It is useful to recall a part of Mostow's structure theorem for
solvmanifolds, as reformulated by Auslander \cite[IV.3]{MR0486307}
and~\cite[p.~271]{MR0486308}:

\begin{theorem}[Mostow, Auslander]
  \label{thm:mostow-auslander}
  Let $D$ be a quasi-lattice in a simply connected, connected,
  solvable Lie group $R$, and let $M(R)=R\rtimes T$ be the semi-simple
  splitting of $R$. Then $T$ is a closed subgroup of
  $\operatorname{Aut}(\mathfrak r)$ and the projection $\pi(D)$ of $D$
  in $T$ is a lattice of $T$.
\end{theorem}

\begin{lemma}
  \label{lem:4:3}
  If the flow $g^\field{R}$ is ergodic and quasi-unipotent on the finite
  volume solvmanifold $R/D$, then the group $R$ is of class (I).
\end{lemma}

\begin{proof}
  We may assume $R$ simply connected and connected and $D$ a
  quasi-lattice. Let $M(R)=N_{M(R)} \rtimes T = R\rtimes T$ be the
  semi-simple splitting of $R$.  Since the one-parameter subgroup
  $g^\field{R}$ is quasi-unipotent, the closure of the projection $\pi(g^\field{R})$
  in the semi-simple factor $T <\operatorname{Aut}(\mathfrak r)$ is a
  compact torus $T' < T$. The surjective homomorphism $\pi:R\to T$
  induces a continuous surjection $R/D \to T/\pi(D)$. By Mostow's
  structure Theorem $R/D $ is compact hence $T/\pi(D)$ is a compact
  torus.  The orbits of $T'$ in $T/\pi(D)$ are finitely covered by
  $T'$. But $g^\field{R}$ acts ergodically on $R/D$, hence $T'$ acts
  ergodically and minimally on $T/\pi(D)$. It follows that $T/\pi(D)$
  consists of a single $T'$ orbit and, since $T', T$ are both
  connected and $T'<T$ and $T' \to T'/\pi(D)$ is a finite cover, we
  obtain that $T'=T$. Thus $T$ consist of quasi-unipotent
  automorphisms of $\mathfrak r$, which implies that for all $g\in R$,
  the automorphism $\mathrm A\mathrm d(g)$ is quasi-unipotent. Hence the group $R$ is
  of class (I).
\end{proof}

The following theorem was first proved in \cite[Thm.\ 4.4]{MR0199308}
under the hypothesis that $D\cap N_{M(R)}= D$. This amounts to suppose
that $D$ is nilpotent, which is the case when $R/D$ supports a minimal
flow, as it is proved in \cite[Thm.~C]{MR0486308}.  A simplification
of the latter proof under the hypothesis that $R/D$ carries an
er\-go\-dic flow appears in \cite[Theorem 7.1]{Starkov}.
\begin{theorem}[Auslander, Starkov]
  \label{thm:auslander-starkov}
  An ergodic flow on a class (I) compact solvable manifold is smoothly
  isomorphic to a nilflow.
\end{theorem}

The first two authors have proved that the main theorem holds for
general nilflows, that is, that the following result holds:
\begin{theorem}[\cite{FF}]
  \label{thm:flaminio-forni}
  An ergodic nilflow which is not toral, has countably many
  independent invariant distributions.
\end{theorem}

In conclusion we have

\begin{proposition}
  \label{prop:4:4}
  An ergodic quasi-unipotent flow on a finite volume solvmanifold is
  either smoothly conjugate to a linear toral flow or it admits
  countably many independent invariant distributions.
\end{proposition}

\section{Partially hyperbolic homogeneous flows}
\label{sec:5}

In the non-compact, finite volume case, by applying results of
D.~Kleinbock and G.~Margulis we are able to generalize
Theorem~\ref{expandingminimal} to flows on \emph{semi-simple}
manifolds.  We think that it is very likely that a general partially
hyperbolic flow on any finite volume manifold has infinitely many
different minimal sets, but we were not able to prove such a general
statement.

For non-compact finite volume we recall the following result by
D.~Kleinbock and G.~Margulis \cite{MR96k:22022}
and its immediate corollary.

\begin{theorem}[Kleinbock and Margulis] \label{thm:kleinbock-margulis}
  Let $G$ be a connected semi-simple Lie group of dimension $n$
  without compact factors, $\Gamma$ an irreducible lattice in $G$. For
  any partially hyperbolic homogeneous flow $g^\field{R}$ on $G/\Gamma$, for
  any closed invariant set $Z\subset G/\Gamma$ of (Haar) measure zero
  and for any nonempty open subset $W$ of $G/\Gamma$, we have that
$$\dim_H(\{x\in W\;|\;g^\field{R} x\;\mbox{is bounded and}\;\overline{g^\field{R} x}\cap Z=\emptyset\})=n.$$
Here $\dim_H$ denotes the Hausdorff dimension.
\end{theorem}

Observe that if the flow $g^\field{R}$ is ergodic, it is enough to assume
that the closed invariant set $Z \subset G/\Gamma$ be proper.

\begin{corollary}\label{cor:kleinbock-margulis}
  Under the conditions of Theorem~\ref{thm:kleinbock-margulis} the
  flow $(G/\Gamma,g^\field{R})$ has infinitely many different compact minimal
  invariant sets.
\end{corollary}

\begin{proposition}\label{prop:5:2}
  Let $G$ be a connected semi-simple Lie group and $G/D$ a finite
  volume space. Assume that the flow $g^\field{R}$ on $G/D$ is partially
  hyperbolic.  Then the flow $(G/\Gamma,g^\field{R})$ has infinitely many
  distinct compact minimal invariant sets, and consequently infinitely
  many $g_\field{R}$-invariant and mutually singular ergodic measures.
\end{proposition}

\begin{proof}
  Let $G=K\times S$ be the decomposition of $G$ as the almost-direct
  product of a compact semi-simple subgroup $K$ and of a totally
  non-compact semi-simple group $S$, with both $K$ and $S$ connected
  normal subgroups. Since the flow $(G/\Gamma,g^\field{R})$ is partially
  hyperbolic $S$ is not trivial. Since $K$ is compact and normal, then
  $D'=DK=KD\subset G$ is a closed subgroup, and since $D\subset KD$,
  then $G/KD$ is of finite volume. Moreover, $(G/K)/DK\sim G/KD$ is of
  finite volume and $G'=G/K\sim S/S\cap K$ is semi-simple without
  compact factor with $D'\subset G'$ a closed subgroup with $G'/D'$ of
  finite volume and a projection $p:G/D\to G'/D'$.  Thus we may assume
  that $G$ is totally non compact, and by
  Proposition~\ref{prop:lattice_red}, that the center of $G$ is finite
  and that $D$ is a lattice.

  If $D$ is irreducible, then the statement follows immediately from
  Corollary~\ref{cor:kleinbock-margulis}.  Otherwise, let $G_i$, for
  $i\in \{1,\dots, l\}$, be connected normal simple subgroups such
  that $G=\prod_iG_i$, $G_i\cap G_j=\{e\}$ if $i\neq j$, and let
  $\Gamma_i=\Gamma\cap G_i$ be an irreducible lattice in $G_i$, for
  each $i\in \{1,\dots, l\}$, with $\Gamma_0=\prod_i \Gamma_i$ of
  finite index in $\Gamma$.  Observe that
  $G/\Gamma_0\sim\prod_iG_i/\Gamma_i$. Let $p:G/\Gamma_0\to G/\Gamma$
  be the finite-to-one covering and let $p_i:G/\Gamma_0\to
  G_i/\Gamma_i$ be the projections onto the factors.  Let $g_0^\field{R}$ be
  the flow induced by the one-parameter group $g^\field{R}$ on $G/\Gamma_0$
  and let $g_i^\field{R}$ be the projected flow on $G_i/\Gamma_i$, for all
  $i\in \{1,\dots, l\}$. Since $\Gamma_i$ is an irreducible lattice in
  $G_i$, whenever $g_i^\field{R}$ is partially hyperbolic we can apply
  Corollary~\ref{cor:kleinbock-margulis}. Since $g_0^\field{R}$ is partially
  hyperbolic there is at least one $j \in \{1, \dots, l\}$ such that
  $g_j^\field{R}$ is partially hyperbolic. By
  Corollary~\ref{cor:kleinbock-margulis}, the flow $g_j^\field{R}$ has a
  countable family $\{K_n \vert n\in \field{N}\}$ of distinct minimal subsets
  of $G_j/\Gamma_j$ such that each $K_n$ supports an invariant
  probability measure $\eta_n$. For all $n\in \field{N}$, let us define $\hat
  \mu_n:=\eta_n\times Leb$ on $G/\Gamma_0$. By construction the
  measures $\hat \mu_n$ are invariant, for all $n\in \field{N}$, and have
  mutually disjoint supports.  Finally, since the map $p:G/\Gamma_0\to
  G/\Gamma$ is finite-to-one, it follows that the family of sets
  $\{p(K_n\times\prod_{i\neq j}G_i/\Gamma_i) \vert n\in \field{N}\}$ consists
  of countably many disjoint closed sets supporting invariant measures
  $ \mu_n:=p_*\hat\mu_n$. The proof of the Proposition is therefore
  complete.
\end{proof}

\section{The general case}
\label{sec:6}

We may now prove our main theorem.

\begin{proof}[Proof of Theorem~\ref{thm:Main}]
  By Remark~\ref{rem:3:1} we may suppose the flow ergodic. Let also
  assume that $G$ is simply connected, by possibly pulling back $D$ to
  the universal cover of $G$. Recall that by Theorem~\ref{thm:3:2} the
  ergodic flow $(G/\Gamma,g^\field{R})$ projects onto the ergodic flow
  $\big(L/\overline{q(D)}, q(g^\field{R})\big)$, where $L$ is the Levi factor
  of $G$ and $q:G\to L$ the projection of $G$ onto this factor. Assume
  that the finite measure space $\big(L/\overline{q(D)}\big)$ is not
  trivial. Then the statement of the theorem follows from
  Proposition~\ref{prop:4:2} if the flow $\big(L/\overline{q(D)},
  q(g^\field{R})\big)$ is quasi-unipotent and by Proposition~\ref{prop:5:2}
  if it is partially hyperbolic.

  If the finite measure space $\big(L/\overline{q(D)}\big)$ is reduced
  to a point, then, using again Theorem~\ref{thm:3:2}, we have $G/D
  \approx R/R\cap D$, where $R$ is the radical of $G$. We obtain in
  this way that our original flow is diffeomorphic to an ergodic flow
  on a finite volume solvmanifold. By Mostow's theorem (see
  Theorem~\ref{thm:mostow}), a finite volume solvmanifold is
  compact. Hence the statement of the theorem follows from
  Theorem~\ref{expandingminimal} if the projected flow is partially
  hyperbolic and by Proposition~\ref{prop:4:4} if it is
  quasi-unipotent. The proof is therefore complete.
\end{proof}

\section{Open problems}
\label{sec:7}

We conclude the paper by stating some (mostly well-known) open
problems and conjectures on the stability and the codimension of
smooth flows.

\begin{conjecture} (A. Katok)
Every homogeneous flow (on a compact
  homogeneous space)
which fails to
  be stable (in the sense that the range of the Lie derivative on the
  space of smooth functions is not closed) projects onto a Liouvillean
  linear flow on a torus. In this case, the flow is still stable on
  the orthogonal complement of the subspace of toral functions (in
  other words, the subspace of all functions with zero average along
  each fiber of the projection).
\end{conjecture}

It is known that hyperbolic and partially hyperbolic, central
isometric (or more generally with uniform sub-exponential central
growth), accessible systems are stable (in the hyperbolic case it
follows by Livshitz theory; in the partially hyperbolic accessible
case see the work of A.~Wilkinson~\cite{Wilkinson} for accessible
partially hyperbolic maps and references therein). In the partially
hyperbolic non-accessible case, several examples are known to be
stable (see \cite{Veech} for toral automorphisms and \cite{Dolgopyat}
for group extensions of Anosov). In the unipotent case, it is proved
in \cite{FF1} that $SL(2,\field{R})$ unipotent flows (horocycles) on finite
volume homogeneous spaces are stable and in \cite{FF} that the above
conjecture holds for nilflows.

\begin{problem} Classify all compact manifolds which admit uniquely
  ergodic flows with $(a)$ a unique invariant distribution (equal to
  the unique invariant measure) up to normalization; $(b)$ a finite
  dimensional space of invariant distributions.
\end{problem}

Example of manifolds (and flows) of type $(a)$ have been found by
A.~Avila, B.~Fayad and A.~Kocsard \cite{AFK}. Note that the Katok
(Greenfield-Wallach) conjecture implies that in all non-toral examples
of type $(a)$ the flow cannot be stable. Recently A.~Avila and
A.~Kocsard \cite{private} have announced that they
have constructed maps on the two-torus having a space of invariant
distributions of arbitrary odd dimension.  It is unclear whether
examples of this type can be stable:

\begin{problem} (M. Herman) Does there exists a stable flow
  with finitely many invariant distributions which is not smoothly conjugate
  to a Diophantine linear flow on a torus?
\end{problem}
The only known example which comes close to an affirmative answer to
this problem is given by generic area-preserving flows on compact
higher genus surfaces \cite{Forni97}, \cite{MMY}.  Such flows are
generically stable and have a finite dimensional space of invariant
distributions in every finite differentiability class (but not in the
class of infinitely differentiable functions).