Let $f$ be a homeomorphism of a topological space onto itself. 
We recall that a minimal subset for $f$ is a closed invariant subset $F$ such that there is no closed invariant subsets of $F$ under $f$. Equivalently, $F$ is a closed subset for which every orbit is dense. Using Zorn's Lemma it can be shown that such sets always exist.


It is easy to see that $R$ cannot be a minimal subset, or that irrationnal rotations on the circle are. 


Is it possible to classify such sets (and/or the associated homeomorphisms), say, in dimension less than two ? 
More precisely, given a homeomorphism of the plane, is it possible to classify its possible minimal subsets ? 


If it turns out that this is too ambitious, are there results of this type for a suitably well-behaved (but still large) class of dynamical systems ? 



Is it realistic to want to classify minimal sub-systems (in small dimension) ?
It is certainly the case that classifying the minimal subsystems of homeomorphisms of compact 2-manifolds presents profound and fundamental difficulties. This is because some very simple transformations, such as analytic diffeomorphisms of the 2-torus, have extremely rich families of minimal sets. 


Let $T \colon X \to X$ be a linear Anosov diffeomorphism of the 2-torus. The topological entropy of $T$ is finite and positive but may be arbitrarily large. If a natural number $k$ is specified, then we may find a linear Anosov diffeomorphism $T$ of the 2-torus $X$ such that the shift transformation on $k$ symbols may be homeomorphically embedded into the dynamical system $(X,T)$ as a compact invariant subset. In particular, every minimal subsystem of the $k$-shift embeds into $(X,T)$ as a minimal subsystem. 


This is problematic because the $k$-shift has an enormous number of minimal subsystems, all of which will be inherited by the Anosov system. Indeed, the combinatorial version of the Jewett-Krieger theorem implies that every ergodic measure-preserving transformation of an abstract probability space which has entropy strictly less than $\log k$ may be embedded into the $k$-shift as a uniquely ergodic minimal subsystem. In particular, for a linear Anosov diffeomorphism of the 2-torus with topological entropy large enough, every ergodic measurable dynamical system with measure-theoretic entropy up to some threshold arises as a uniquely ergodic minimal subsystem.


This already presents us with an enormous number of minimal subsystems, because for each $h \geq 0$ there exist uncountably many ergodic measurable dynamical systems of entropy $h$ which are not pairwise equivalent. This is then compounded by the fact that some minimal systems of $(X,T)$ will not arise from such an embedding, the fact that the embedding of the abstract ergodic system into the $k$-shift is in general not unique, the fact that the embedding of the $k$-shift into $(X,T)$ is in general not unique, and the fact that the $k$-shift itself has additional minimal subsystems. Indeed, there is a further theorem due to Denker, Grillenberger and Sigmund which implies that for any finite collection of abstract ergodic transformations all having entropy strictly below $\log k$, we can find a minimal subsystem of the $k$-shift which has an embedded copy of each of these transformations as its only ergodic measures.


On the basis of the above considerations I think that a satisfactory classification of the minimal subsystems of homeomorphisms of the 2-torus is improbable!


