You are an expert at summarizing long articles. Proceed to summarize the following text: 
a comprehensive understanding of hamiltonian dynamics is a long outstanding problem in nonlinear and statistical physics , which has important applications in various other areas of physics . typical hamiltonian systems are nonhyperbolic as they exhibit mixed phase space with coexisting regular and chaotic regions . over the past years , a number of ground - breaking works @xcite have increasingly elucidated the asymptotic behavior of such systems and it is now well understood that , because of the stickiness due to kolmogorov - arnold - moser ( kam ) tori , the chaotic dynamics of typical hamiltonian systems is fundamentally different from that of hyperbolic , fully chaotic systems . here `` asymptotic '' means in the limit of large time scales and small length scales . but in realistic situations , the time and length scales are limited . in the case of hyperbolic systems , this is not a constraint because the ( statistical ) self - similarity of the underlying invariant sets guarantees the fast convergence of the dynamical invariants ( entropies , lyapunov exponents , fractal dimensions , escape rates , etc ) and the asymptotic dynamics turns out to be a very good approximation of the dynamics at finite scales . in nonhyperbolic systems , however , the self - similarity is usually lost because the invariant sets are not statistically invariant under magnifications . as a result , the finite - scale behavior of a hamiltonian system may be fundamentally different from the asymptotic behavior considered previously , which is in turn hard to come by either numerically @xcite or experimentally @xcite . the aim of this paper is to study the dynamics of hamiltonian systems at finite , physically relevant scales . to the best of our knowledge , this problem has not been considered before . herewith we focus on hamiltonian chaotic scattering , which is one of the most prevalent manifestations of chaos in open systems , with examples ranging from fluid dynamics @xcite to solid - state physics @xcite to general relativity @xcite . we show that the finite - scale dynamics of a hamiltonian system is characterized by _ effective _ dynamical invariants ( e.g. , effective fractal dimension ) , which : ( 1 ) may be significantly different from the corresponding invariants of the asymptotic dynamics ; ( 2 ) depend on the resolution but can be regarded as constants over many decades in a given region of the phase space ; and ( 3 ) may change drastically from one region to another of the _ same _ dynamically connected ( ergodic ) component . these features are associated with the slow and nonuniform convergence of the invariant measure due to the breakdown of self - similarity in nonhyperbolic systems . to illustrate the mechanism behind the properties of the effective invariants , we introduce a simple deterministic model which we build on the observation that a hamiltonian system can be represented as a chain of hyperbolic systems . the paper is organized as follows . we start , in sec . [ s2 ] , with the analysis of the invariant measure and the outline of the transport structures underlying its convergence . our chain model is introduced and analyzed in sec . [ 3 ] . the effective fractal dimension is defined in sec . [ 4 ] and its properties are verified for a specific system in sec . [ 5 ] . conclusions are presented in the last section . for concreteness , consider a two - dimensional area preserving map with a major kam island surrounded by a chaotic region . one such map captures all the main properties of a wide class of hamiltonian systems with mixed phase space . when the system is open ( scattering ) , almost all particles initialized in the chaotic region eventually escape to infinity . we first study this case with a diffusive model for the transversal motion close to the main kam island , obtaining an analytical expression for the probability density @xmath0 of particles remaining in the scattering region at time @xmath1 and distance @xmath2 from the island [ see appendix ] . we find that , in the case of chaotic scattering , a singularity develops and the invariant measure , given by @xmath3 , accumulates on the outermost kam torus of the kam island [ appendix ] . physically , this corresponds to the tendency of nonescaping particles to concentrate around the regular regions . dynamically , the stickiness due to kam tori underlies two major features of hamiltonian chaotic scattering , namely the algebraic decay of the survival probability of particles in the scattering region @xcite and the integer dimension of the chaotic saddle @xcite , and distinguishes this phenomenon from the hyperbolic chaotic scattering characterized by exponential decay and noninteger fractal dimension . however , the convergence of the measure is rather slow and highly nonuniform , as shown in fig . [ fig1 ] for typical parameters , which is in sharp contrast with the fast , uniform convergence observed in hyperbolic systems . our main results are ultimately related to this slow and nonuniform convergence of the invariant measure . previous works on transport in hamiltonian systems have used stochastic models , where invariant structures around kam islands are smoothened out and the dynamics is given entirely in terms of a diffusion equation @xcite or a set of transition probabilities ( markov chains or trees ) @xcite . the stochastic approach is suitable to describe transport properties ( as above ) , but can not be used to predict the behavior of dynamical invariants such as lyapunov exponents and fractal dimensions . here we adopt a deterministic approach where we use the cantori surrounding the kam islands to split the nonhyperbolic dynamics of the hamiltonian system into a chain of hyperbolic dynamical systems . cantori are invariant structures that determine the transversal transport close to the kam islands @xcite . there is a hierarchy of infinitely many cantori around each island . let @xmath4 denote the area of the scattering region outside the outermost cantorus , @xmath5 denote the annular area in between the first and second cantorus , and so on . as @xmath6 is increased , @xmath7 becomes thinner and approaches the corresponding island . for simplicity , we consider that there is a single island @xcite and that , in each iteration , a particle in @xmath7 may either move to the outer level @xmath8 or the inner level @xmath9 or stay in the same level @xcite . let @xmath10 and @xmath11 denote the transition probabilities from level @xmath6 to @xmath12 and @xmath13 , respectively . a particle in @xmath4 may also leave the scattering region , and in this case we consider that the particle has escaped . the escaping region is denoted by @xmath14 . the chaotic saddle is expected to have points in @xmath7 for all @xmath15 . it is natural to assume that the transition probabilities @xmath10 and @xmath11 are constant in time . this means that each individual level can be regarded as a hyperbolic scattering system , with its characteristic exponential decay and noninteger chaotic saddle dimension . therefore , a nonhyperbolic scattering is in many respects similar to a sequence of hyperbolic scatterings . we now introduce a simple deterministic model that incorporates the above elements and reproduces essential features of the hamiltonian dynamics . our model is depicted in fig . [ fig2 ] and consists of a semi - infinite chain of 1-dimensional `` @xmath16-shaped '' maps , defined as follows : @xmath17 where @xmath18 and @xmath19 ( @xmath20 ) . if @xmath2 falls in the interval @xmath21 , where @xmath22 is not defined , the `` particle '' is considered to have crossed a cantorus to the `` outer level '' @xmath12 . this interval is mapped uniformly to @xmath23 $ ] , and the iteration proceeds through @xmath24 . symbolically , this is indicated by @xmath25 . similarly , if @xmath2 falls into @xmath26 , where @xmath27 , the particle goes to the `` inner level '' , and @xmath28 . particles that reach @xmath29 are considered to have escaped . the domain of @xmath22 is denoted by @xmath30 and is analogous to @xmath7 in a hamiltonian system , where @xmath10 and @xmath11 represent the transition probabilities . the transition rate ratios @xmath31 and @xmath32 are taken in the interval @xmath33 and are set to be independent of @xmath6 , where @xmath34 . the parameter @xmath35 is a measure of the fraction of particles in a level @xmath6 that will move to the inner level @xmath13 when leaving level @xmath6 , while @xmath36 is a measure of how much longer it takes for the particles in the inner level to escape . the nondependence on @xmath6 corresponds to the approximate scaling of the cantori suggested by the renormalization theory @xcite . despite the hyperbolicity of each map , the entire chain behaves as a nonhyperbolic system . for a uniform initial distribution in @xmath37 , it is not difficult to show @xcite that the number of particles remaining in the chain after a long time @xmath1 decays algebraically as @xmath38 , and that the initial conditions of never escaping particles form a zero lebesgue measure fractal set with box - counting dimension 1 . however , the finite - scale behavior may deviate considerably from these asymptotics , as shown in fig . [ fig3 ] .    in fig . [ fig3](a ) we show the survival probability @xmath39 as a function of time . for small @xmath35 and @xmath36 , the curve is composed of a discrete sequence of exponentials with scaling exponents @xmath40 , which decrease ( in absolute value ) as we go forward in the sequence . the length of each exponential segment is of the order of @xmath35 in the decay of @xmath39 and @xmath41 in the variation of @xmath42 . this striking behavior is related to the time evolution of the density of particles inside the chain . this is shown in fig . [ fig3](b ) , where we plot the average position @xmath43 of an ensemble of particles initialized in @xmath37 ( i.e. , @xmath44 ) . the transitions between successive exponentials in the decay of @xmath39 [ fig . [ fig3](a ) ] match the transitions from a level @xmath6 to the next in the average position of the remaining particles [ fig . [ fig3](b ) ] . in a hamiltonian system , the increase of @xmath43 in time is related to the development of the singular invariant measure anticipated in our diffusion analysis [ see fig .  [ fig1 ] ] . the piecewise exponential behavior of @xmath39 is smoothened out for large @xmath35 and @xmath45 [ figs . [ fig3](a ) and [ fig3](b ) ] .    in fig . [ fig3](c ) we show the fractal dimension of the set of initial conditions of never escaping particles as computed from the uncertainty algorithm @xcite , which consists in measuring the scaling of the fraction @xmath46 of _ @xmath47-uncertain _ points ( initial points whose escaping time is different from the escaping time of points taken @xmath47 apart ) . the scaling is statistically well defined over decades and the exponent @xmath48 can be computed accurately . however , the resulting dimension @xmath49 is not only significantly smaller than 1 but also depends critically on the region @xmath50 of the phase space where it is computed . the convergence of the dimension is indeed so slow that it can only be noticed when observed over very many decades of resolution , as shown in fig . [ fig3](d ) where data of fig . [ fig3](c ) is plotted over 35 decades ! initially smaller , the dimension measured for @xmath51 approaches the dimension measured for @xmath52 as the scale @xmath47 is reduced beyond @xmath53 ( i.e. the corresponding curves in fig .  [ fig3](d ) become parallel ) . as shown in fig . [ fig3](d ) , this behavior is related to a transition in the average innermost level @xmath54 reached by the particles launched from @xmath47-uncertain points . as @xmath47 is further reduced , new transitions are expected . the dimension measured in between transitions is mainly determined by the dimension @xmath55 , @xmath56 , of the corresponding element of the chain . for given @xmath6 and @xmath47 , the measured dimension is larger when @xmath50 is taken in a denser part of the invariant set , such as in the subinterval of @xmath37 first mapped into @xmath57 [ fig . [ fig3](c ) ; diamonds ] , because @xmath58 is larger in these regions . in some regions , however , the measured dimension is quite different from the asymptotic value even at scales as small as @xmath59 . this slow convergence of the dimension is due to the slow increase of @xmath58 , which in a hamiltonian system is related to the slow convergence of the invariant measure [ fig . [ fig1 ] ] . the convergence is even slower for smaller @xmath35 and larger @xmath36 . incidentally , the experimental measurements of the fractal dimension are usually based on scalings over less than two decades @xcite . therefore , at realistic scales the dynamics is clearly not governed by the asymptotic dynamical invariants . our results on the chain model motivate us to introduce the concept of effective dynamical invariants . as a specific example , we consider the _ effective _ fractal dimension , which , for the intersection of a fractal set @xmath60 with a @xmath61-dimensional region @xmath50 , we define as @xmath62 where @xmath63 , and @xmath64 and @xmath65 are the number of cubes of edge length @xmath66 needed to cover @xmath67 and @xmath50 , respectively @xcite . we take @xmath50 to be a generic segment of line [ i.e. , @xmath68 in eq . ( [ 2 ] ) ] intersected by @xmath60 on a fractal set . in the limit @xmath69 , we recover the usual box - counting dimension @xmath70 of the fractal set @xmath67 , which is known to be 1 for all our choices of @xmath50 . however , for any practical purpose , the parameter @xmath47 is limited and can not be made arbitrarily small ( e.g. , it can not be smaller than the size of the particles , the resolution of the experiment , and the length scales neglected in modeling the system ) . at scale @xmath47 the system behaves as if the fractal dimension were @xmath71 ( therefore `` effective '' dimension ) . in particular , the final state sensitivity of particles launched from @xmath50 , with the initial conditions known within accuracy @xmath72 , is determined by @xmath73 rather than @xmath74 : as @xmath47 is variated around @xmath72 , the fraction of particles whose final state is uncertain scales as @xmath75 , which is different from the prediction @xmath76 . this is important in this context because , as shown in fig . [ fig3 ] ( where the effective dimension is given by @xmath49 ) , the value of @xmath71 may be significantly different from the asymptotic value @xmath77 even for unrealistically small @xmath47 and may also depend on the region of the phase space . similar considerations apply to many other invariants as well . we now return to the hamiltonian case . consider a scattering process in which particles are launched from a line @xmath50 transversal to the stable manifold @xmath78 of the chaotic saddle . based on the construction suggested by the chain model , it is not difficult to see that @xmath79 exhibits a hierarchical structure which is not self - similar and is composed of infinitely many nested cantor sets , each of which is associated with the dynamics inside one of the regions @xmath7 . as a consequence , the effective dimension @xmath71 in hamiltonian systems is expected to behave similarly to the effective dimension in the chain model [ figs . [ fig3](c ) and [ fig3](d ) ] . in particular , @xmath71 is expected to display a strong dependence on @xmath50 and a weak dependence on @xmath47 . we test our predictions on the area preserving hnon map : @xmath80 , where @xmath81 is the bifurcation parameter . in this system , typical points outside kam islands are eventually mapped to infinity . because of the symmetry @xmath82 , where @xmath83 , the stable and unstable manifolds of the chaotic saddle are obtained from each other by exchanging @xmath2 and @xmath84 . for @xmath85 , the system displays a period - one and a period - four major island , as shown in fig . [ fig4](a ) . in the same figure we also show the complex invariant structure around the islands , the stable manifold of the chaotic saddle , and three different choices for the line of starting points : a large interval away from the islands ( @xmath86 ) , a small subinterval of this interval where the stable manifold appears to be denser ( @xmath87 ) , and an interval closer to the islands ( @xmath88 ) . the corresponding effective dimensions are computed for a wide interval of @xmath47 . the results are shown in fig . [ fig4](b ) : @xmath89 , @xmath90 , and @xmath91 for @xmath92 . these results agree with our predictions that the effective fractal dimension has the following properties : @xmath93 may be significantly different from the asymptotic value @xmath94 of the fractal dimension ; @xmath93 depends on the resolution @xmath47 but is nearly constant over decades ; @xmath93 depends on the region of the phase space under consideration and , in particular , is larger in regions closer to the islands and in regions where the stable manifold is denser . similar results are expected for any typical hamiltonian system with mixed phase space . we have shown that the finite - scale dynamics of hamiltonian systems , relevant for realistic situations , is governed by effective dynamical invariants . the effective invariants are not only different from the asymptotic invariants but also from the usual hyperbolic invariants because they strongly depend on the region of the phase space . our results are generic and expected to meet many practical applications . in particular , our results are expected to be relevant for fluid flows , where the advection dynamics of tracer particles is often hamiltonian @xcite . in this context , a slow nonuniform convergence of effective invariants is expected not only for time - periodic flows , capable of holding kam tori , but also for a wide class of time - irregular incompressible flows with nonslip obstacles or aperiodically moving vortices .        the diffusion model is : @xmath95 $ ] , where @xmath96 is the probability density of all particles , @xmath97 , and @xmath98 @xcite . the outermost torus of the kam island is at @xmath99 , where the diffusion rate ( proportional to @xmath100 ) vanishes . in a chaotic scattering process the initial distribution of particles is localized apart from the confining islands . we take @xmath101 , @xmath102 , and consider a particle to escape when it reaches @xmath103 . under the approximation that for large @xmath104 the return of particles can be neglected , we disregard the boundary condition @xmath105 and we take the solution to be the corresponding green function : @xmath106 , where @xmath107 , @xmath108 is @xmath84 at @xmath109 , @xmath110 , and @xmath111 is the modified bessel function , which scales as @xmath112 for small @xmath113 @xcite . for any fixed @xmath114 , we can show that the distribution for large @xmath1 decreases as @xmath115 , where @xmath110 . on the other hand , as shown in ref . @xcite , the fraction of particles in the interval @xmath116 decays algebraically as @xmath117d@xmath118 . combining these two results , it follows that the normalized probability density @xmath119 decreases as @xmath120 at each fixed @xmath121 for large enough @xmath1 and diverges arbitrarily close to @xmath99 . chain models have also been used to describe power laws generated by a hierarchy of repellers in the motion towards the feigenbaum attractor [ p. grassberger and m. scheunert , j. stat . * 26 * , 697 ( 1981 ) ] and to study deterministic diffusion [ r. klages and j. r. dorfman , phys lett . * 74 * , 387 ( 1995 ) ] . the escape rate follows from the same argument used in [ j. d. meiss and e. ott , physica d * 20 * , 387 ( 1986 ) ] . the fractal dimension follows from the observation that the dimension of the invariant set of map @xmath22 goes to @xmath94 as @xmath122 . lai , m. ding , c. grebogi , and r. blmel , phys . a * 46 * , 4661 ( 1992 ) ; w. breymann , z. kovcs , and t. tl , phys . e * 50 * , 1994 ( 1994 ) ; p. gaspard and j. r. dorfman , phys . e * 52 * , 3525 ( 1995 ) ; v. constantoudis and c. a. nicolaides , phys . e * 64 * 056211 ( 2001 ) .
an adequate characterization of the dynamics of hamiltonian systems at physically relevant scales has been largely lacking . 
 here we investigate this fundamental problem and we show that the finite - scale hamiltonian dynamics is governed by effective dynamical invariants , which are significantly different from the dynamical invariants that describe the asymptotic hamiltonian dynamics . 
 the effective invariants depend both on the scale of resolution and the region of the phase space under consideration , and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the hamiltonian system is modeled as a chain of hyperbolic systems . 
 [ phys . 
 rev . 
 e * 71 * , 036215 ( 2005 ) ]