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## Alpern genericity theorem

Frédéric Le Roux has written a very lucid exposition of the Alpern genericity theorem. This theorem states that the same ergodic properties are generic in the set $Auto(T^d)$ of volume-preserving measurable transformations and in the set $Homeo(T^d)$ of volume-preserving  homeomorphisms.

More precisely,  endow $Auto(T^d)$   with the weak topology, i.e., the coarsest generated by $\mu\mapsto\mu(A)$, for all measurable subsets $A$ and equip $Homeo(T^d)$  with the uniform distance.

The theorem considers any $P\subset Auto(T^d)$  invariant under volume-preserving measurable isomorphisms. It states that $P$  is a dense $G_\delta$ subset of $Auto(T^d)$ if and only if $P\cap Homeo(T^d)$ is itself a dense $G_\delta$ subset of $Homeo(T^d)$.

## Orders of accumulation of entropy structures

Kevin McGoff gave a talk at Orsay on his work on the theory of entropy srtuctures and symbolic extensions. This theory was founded by Mike Boyle and Tomasz Downarowicz, among others.

This theory relates the continuity properties of the measure-theoretic entropy function with the existence of symbolic topological extensions with measure-theoretic entropies as close as possible to those of the initial system. This theory ascribes to any topological dynamical system an order of accumulation. M. Boyle and T. Downarowicz have shown that this is a countable ordinal.

David Burguet and K. McGoff have shown that any countable ordinal can be achieved by some topological dynamics. The proof relies on a realization theorem of T. Downarowicz and S. Serafin.

K. McGoff explained how he was able, by a more precise and direct construction to achieve the same on any prescribed compact manifold. The transformation can be chosen to be homeomorphic if the dimension is 2 or more.

## C^2 surface diffeomorphisms always have a symbolic extension

Most of topological dynamics studies systems of the form $T:X\to X$ where $T$ is a continuous self-map and $X$ is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., $\sigma:S\to X$ where $S$ is a closed subset of $\{1,\dots,d\}^{\mathbb Z}$ and $\sigma((x_n)_{n\in\mathbb Z})=(x_{n+1})_{n\in\mathbb Z}$ such that $\sigma(S)=S$.

J. Auslander asked about the obstructions for a topological system $T:X\to X$ to have a symbolic extension, i.e., a symbolic system $\sigma:S\to S$ and a continuous surjection $\pi:S\to X$ commuting with the dynamics: $\pi\circ\sigma =T\circ\pi$. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?

M. Boyle showed that this was indeed the case. With D. Fiebig and U. Fiebig, he showed that asymptotically h-expansive systems (including $C^\infty$ self-maps of compact manifolds by a result of mine based on Yomdin’s theory) always have a “nice” symbolic extension. T. Downarowicz and S. Newhouse showed that generic $C^1$ map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.

T. Downarowicz and A. Maas showed that $C^r$ interval maps also always have symbolic extensions for $1.

David BURGUET has finally proved the same for arbitrary $C^2$ surface diffeomorphisms, see his preprint here.

Behind these works there is a rich and beautiful topological/ergodic/functional-analytical theory of entropy (called the entropy structure by T. Downarowicz) which does yet have the audience it deserves, in my opinion.

## Sylvain CROVISIER’s HDR

On November 25th, 2009, Sylvain CROVISIER defended his habilitation à diriger des recherches titled Perturbation de la dynamique de difféomorphismes en petite régularité. He first explained basic perturbation techniques:

• the Anosov-Katok procedure: you use more and more distorted conjugacies such that the limiting dynamics has new properties;
• the closing lemma of Pugh and the subsequent connecting lemmas of Hayashi and Bonatti-Crovisier: you try to glue pieces of orbits while avoiding intermediate visits to the support of the pertubation. This may force you to drop pieces of the orbits. Crovisier nevertheless managed to prove a generalized shadowing lemma constructing orbits guaranteed to visit some neighborhoods.

He then explained how the above was put to work. Together with François Béguin and Frédéric Le Roux, he used the first tool to realize almost arbitrary measurable dynamics as minimal, uniquely ergodic homeomorphisms on arbitray compact manifolds.

With Christian Bonatti, Enrique Pujals and other co-workers, he used the second set of tools (together with constructions of dominated splittings by Wen, Liao-Wen and his own central models) to get deep new results on the dynamics of C^1 generic diffeomorphisms.

Some of the most important ones state that, after a small C^1 perturbation, any diffeomorphism either has a strong global structure (i.e., a phenomenon in the sense of Pujals) or presents a simple obstruction (a mechanism, which one would like to be robust). Namely, up to C^1 perturbations, any diffeomorphism of a compact manifold is:

• Morse-Smale unless it has a  transverse homoclinic intersection (this provides a description of an open and dense subset of the C^1 diffeomorphisms called the weak Palis conjecture);
• partially hyperbolic with a central bundle which is one-dimensional or a sum of two one-dimensional sub-bundles, unless it has a heterodimensional cycle or a homoclinic tangency;
• essentially hyperbolic (hyperbolic from the point of view of its attractors and repellers), unless it has a heterodimensional cycle or a homoclinic tangency.

• It is true that Mañé already characterized the non-hyperbolic diffeomorphisms as those having periodic points that can be made to bifurcate but the goal here is to get robust obstructions;
• It is not clear that dynamics with infinitely many chain recurrent classes can exist robustly: there is no known mechanism for that;
• It seems very difficult to get beyond C^1 with anything like the current techniques – even C^1+1/log seems out of reach;
• It is a reasonable to question to try and develop more precise description of the dynamics especially for the situations that hold on C^1 open sets and therefore occur on C^2 open sets, a usual requisite of ergodic theory techniques;
• The techniques have yielded results very analoguous to Zeeman Tolerance Stability conjecture. However such kinds of stability seem to have more philosophical appeal than a true rôle in the mathematical theory, as opposed to structural stability.
• It is not clear that the above decomposition of the space of C^1 diffeomorphisms extend to higher smoothness (even disregarding the enormous technical difficulties pertaining to the closing lemmas). For instance Bonatti and Diaz hope to show that any diffeomorphism can be C^1 approximated by a hyperbolic one or by one with a homoclinic tangency but this is not the case in the C^2 topology, because of Newhouse phenomenon.
• The techniques seem insufficient to study prevalent or Kolmogorov-typical dynamics.