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## Workshop “Equilibrium states for dynamical systems arising from Geometry

In July, I took part in a workshop on the thermodynamical formalism for dynamical systems of geometrical origin. The workshop was organized by Keith Burns, Vaughn Climenhaga, Todd Fisher, and Dan Thompson at the AIM in San Jose California. The format of such workshops is dedicated to collaborations rather than formal lectures.

I gave a lecture on “Symbolic dynamics and spectral decomposition”. The other lectures:

• Equilibrium states and non-uniform specification by Vaughn Climenhaga
• Equilibrium states for geodesic flow in non-positive curvature by Dan Thompson
• Teichmüller geodesic flow and Rauzy induction by Jayadev Athreya
• Equilibrium states via geometric measure theory by Agnieszka Zelerowicz
• Anisotropic Banach spaces for dispersing billiards by Mark Demers
• Geodesic flow for CAT(0) and CAT(-1) spaces by Dave Constantine
• Symbolic dynamics for non-uniform and hyperbolic systems by Yuri Lima
• Flexibility of entropies in a fixed conformal class by Alena Erchenko
• Rigidity of abelian actions and interactions with Gibbs states by Kurt Vinhage

Videos of lectures, etc. are available from the above page.

## Renaud Leplaideur: Renormalisation des potentiels et transitions de phase

Renaud Leplaideur a exposé au groupe de travail de théorie ergodique ses tout derniers travaux avec H. Bruin, A.T. Baraviera et A.O. Lopes. Ils ont en particulier construit, pour tout $0, une application $f_a:S^1\to S^1$ de classe $C^1$ présentant une transition de phase et un compact $K$, invariant, uniquement ergodique et indifférent: (i) $f_a'\geq 1$; (ii) $(f_a')^{-1}(1)=K$; (iii) le potentiel $- \log f_a'$ possède plusieurs mesures d’équilibre.

Ceci généralise l’exemple bien connu de Manneville-Pomeau. Plusieurs autres résultats suggèrent des liens entre cette non-unicité et une certaine invariance par renormalisation du potentiel, sans qu’il soit clair qu’il s’agisse là d’un phénomène général ou encore d’un analogue mathématique du lien entre phénomène critique et invariance par renormalisation établi et exploité par les physiciens théoriciens depuis Wilson et les années 1970.

## Cellular Automata Modeling Reliable Computers: 3D

Real-world computers make mistakes, in the sense that once in a while an instruction is executed incorrectly, perhaps because of a corrupted disk. One could naively think that, given, a maximum acceptable probability of an incorrect final result, this would impose a bound on the complexity of possible computation or require an exponential number of repetitions. However (and similarly to the central result of Shannon’s information theory), one can do much better as was explained by Péter Gàcs in his mini-course in Marseilles. P. Gàcs slides can be found here.

Computers are modelized as probabilistic cellular automata: the new states (indexed by $\mathbb Z^d$ are independent conditioned on the old states and each follows a law which is a fixed function of the old states in a neighborhood.  These local transitions are assumed to be “noisy”, i.e., all states have positive probability.

Remark. This “noisiness” does not imply ergodicity (in the sense of Markov chains, i.e., there is a unique stationary probability measure), which is fortunate since ergodicity implies that the initial data is eventually forgotten!

Question. When $d=2$, the voting model is expected to be non-ergodic but there is a proof only for a continuous time version with specific parameters that can be related to the Ising model.

It is observed that one-dimensional cellular automata cannot compute reliably in the presence of noise. In a way, there is not enough long range communication for cells on the boundary of an erroneous island to tell on which side is the island… The main result of the first lecture was the following:

Theorem (3D-simulation with infinite redundancy). Let U be some one-dimensional cellular automaton. Then there is a 3-dimensional cellular automaton V and z constant C such that, if the local transitions are noisy but with sufficiently small error probability $\epsilon$, the probability that a given V-state at site $(i,j,k)$ at time $n$ is different from the U-state at site $i$ at the same time is bounded by $C\epsilon$.

There is a version of this result with finite redundancy. Specifically, for a computation which requires a space $S$ and a time $T$ and a maximal error probability at a given site of $\delta>0$, one can replace the infinite extension $\mathbb Z\times\mathbb Z^2$ by a finite one $\{0,\dots,S\}\times \times\{0,\dots,N\}^2$ where $N=\mathcal O(\log(ST)/\delta)$.

The proof relies on a decomposition of the occurence of the “faults” in a hierarchical structures (at level 0, one has only distant single faults, at level 1, one also allows more distant small balls containing faults, etc.).

The second lecture, dealing with reliable computations in 2D, will be reviewed in the following post.

## 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.

## Dilation factors of pseudo-Anosov homeomorphisms

Erwan Lanneau gave a talk in Orsay about his work on surfaces of translation. Pseudo-Anosov homeomorphisms are homeomorphisms of such surfaces which are affine away from the singular set of the surface and whose differential is hyperbolic. The dilation factor is the dominant eigenvalue of that differential. It is a Perron number.

The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that $c_1/g\leq \log \delta(g)\leq c_2/g$.

The main result of the talk is that the above does not hold when restricted to a given type of translation surfaces. More precisely, the moduli space of translation surfaces of given genus splits into connected components (at most three), one of them corresponding to hyperellipticity and the following holds:

Theorem (Boissy-Lanneau) Let $\Phi$ be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with $2g+2$ fixed points). Assume that $\Phi$ has a unique singularity. Then its dilation is strictly greater than $\sqrt{2}$ (but approach this value as $g\to\infty$).

## 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.