Surface Homeomorphisms

A piecewise affine surface homeomorphism is a homeomorphism of a topological space admitting a finite triangulation whose simplicial complex is a homogeneous 2-complex.

These transformations can be considered as toy models for surface diffeomorphisms with distortion replaced by singularities which are localized. As for surface diffeomorphisms, non zero entropy implies nonuniform hyperbolicity.

Some important sub-classes: Lozi family

# Entropy theory

There is a simple combinatorial expression for topological entropy (similar to the Misiurewicz-Slenk formula for piecewise monotone maps). The natural partition $P$ is the collection of the open 2-simplices in the canonical triangulation for $f$. A $P$-cylinder of order $n$ is a non-empty intersection of the form $A_0\cap f^{-1}A_1\cap\dots\cap f^{-n+1}A_{n-1}$ where $A_i\in P$.

Theorem 1 (Ishii-Sands [5]) Let $f$ be a piecewise affine surface homeomorphism. Let $N(f,P,n)$ be the number of $P$-cylinders of order $n$. Then $h_{top}(f)=\limsup_{n\to\infty} (1/n)\log N(f,P,n)$.

This formula can be proved using the symbolic dynamics $\Sigma(f,P)$ naturally defined by $P$ for any PA transformation and checking that for a piecewise surface homeomorphism $f$, $\Sigma(f,P)$ has the same topological entropy as $f$.

In higher dimensions or in the non-invertible setting this formula does not hold, see [2],[6]

Theorem 2 (Buzzi [3]) Consider a piecewise affine surface homeomorphism. If it has non zero topological entropy, then there is a non-zero, finite number of ergodic measures with maximal entropy. Each such measure is a finite extension of a Bernoulli measure.

It is a conjecture that this holds for $C^\infty$ smooth surface diffeomorphisms.

The proof of the above theorem yields:

Proposition 3 (Buzzi [3]). Let $f$ be a piecewise affine surface homeomorphism. For any $\epsilon>0$ there exists a compact invariant set $K$ disjoint from the singularities of $f$ such that $h_{top}(f|K)>h_{top}(f)-\epsilon$.
In particular, topological entropy is lower semi-continuous.

## Problems

• Extend all of the above to more general dynamics…
• Analyze more precisely the dynamics (e.g., define and study zeta functions).
• Prove Theorem 2 by defining a natural class of symbolic dynamics, showing that such dynamics satisfy the conclusion of Theorem 2 and checking that the symbolic dynamics of piecewise affine surface homeomorphisms belong to that class.
• State and prove a Pruning Front Conjecture [4] for piecewise affine surface homeomorphisms, that is, find an "effective" way in which symbolic information about the dynamics of the singularity lines determines the symbolic dynamics of the whole system.
• Prove or disprove the upper semi-continuity of the topological entropy.

# Physical measures

Remark. The results and problems quoted here are not known to be specific of dimension 2 though the situation for piecewise expanding PA maps suggests that higher dimensions might be much more difficult.

Assuming the existence of globally smooth stable or unstable foliation and additional technical conditions, existence of a complete system of physical measures has been proved by Baladi and Gouezel [1] by using a suitable Banach space (their result is not restricted to PA dynamics or to two dimensions).

They dealt in particular with hyperbolic matrices modulo 1: self-maps of the cube of the form $x\mapsto Ax\mod\mathbb Z^d$ where $d\geq1$ and $A$ is an affine map whose linear part is invertible with no eigenvalue on the unit circle. Their approach yields a wealth of information about these measures. We refer to their paper for reference to previous works.

## Problems

• Prove that all (or almost all) piecewise uniformly hyperbolic PA surface homeomorphisms have a complete finite system of physical measures.
• Prove or disprove that for all PA surface homeomorphisms, for almost every initial condition and all continuous functions, the ergodic averages converge.
Bibliography
1. Viviane Baladi, Sebastien Gouezel, Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. IHP (to appear).
2. Jerome Buzzi, Intrinsic ergodicity of affine maps on $[0,1]^d$, Monat. Math. 124 (1997), no. 2, 97-118.
3. Jerome Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergod. th. dynam. systems (to appear) - preprint on arxiv.
4. Yutaka Ishii, Towards a kneading theory for Lozi mappings. I. A solution of the pruning front conjecture and the first tangency problem, Nonlinearity 10 (1997), no. 3, 731-747.
5. Yutaka Ishii, Duncan Sands, Lap number entropy formula for piecewise affine and projective maps in several dimensions, Nonlinearity 20 (2007), 2755-2772.
6. Boris Kruglikov, Martin Rypdal, Entropy via multiplicity, Discrete Contin. Dyn. Syst. 16 (2006), no. 2, 395-410.
page revision: 8, last edited: 23 Mar 2009 12:28