American Institute of Mathematical Sciences

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Hölder-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms.

As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinaï billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a Hölder-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.

We study joining rigidity in the class of von Neumann flows with one singularity. They are given by a smooth vector field \begin{document}$ \mathscr{X} $\end{document} on \begin{document}$ \mathbb{T}^2\setminus \{a\} $\end{document}, where \begin{document}$ \mathscr{X} $\end{document} is not defined at \begin{document}$ a\in \mathbb{T}^2 $\end{document} and \begin{document}$ \mathscr{X} $\end{document} has one critical point which is a center. It follows that the phase space can be decomposed into a (topological disc) \begin{document}$ D_\mathscr{X} $\end{document} and an ergodic component \begin{document}$ E_\mathscr{X} = \mathbb{T}^2\setminus D_\mathscr{X} $\end{document}. Let \begin{document}$ \omega_\mathscr{X} $\end{document} be the 1-form associated to \begin{document}$ \mathscr{X} $\end{document}. We show that if \begin{document}$ |\int_{E_{\mathscr{X}_1}}d\omega_{\mathscr{X}_1}|\neq |\int_{E_{\mathscr{X}_2}}d\omega_{\mathscr{X}_2}| $\end{document}, then the corresponding flows \begin{document}$ (v_t^{\mathscr{X}_1}) $\end{document} and \begin{document}$ (v_t^{\mathscr{X}_2}) $\end{document} are disjoint. It also follows that for every \begin{document}$ \mathscr{X} $\end{document} there is a uniquely associated frequency \begin{document}$ \alpha = \alpha_{\mathscr{X}}\in \mathbb{T} $\end{document}. We show that for a full measure set of \begin{document}$ \alpha\in \mathbb{T} $\end{document} the class of smooth time changes of \begin{document}$ (v_t^\mathscr{X_ \alpha}) $\end{document} is joining rigid, i.e., every two smooth time changes are either cohomologous or disjoint. This gives a natural class of flows for which the answer to [15,Problem 3] is positive.

We study the near action of the group \begin{document}$ \mathrm{PC} $\end{document} of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of \begin{document}$ \mathrm{PC} $\end{document} is said to be realizable if it can be lifted to a group of permutations of the circle.

We prove that every finitely generated abelian subgroup of \begin{document}$ \mathrm{PC} $\end{document} is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips.

The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.