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.

We show that the number of square-tiled surfaces of genus \begin{document}$ g $\end{document}, with \begin{document}$ n $\end{document} marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most \begin{document}$ L $\end{document} squares, is asymptotic to \begin{document}$ L^{6g-6+2n} $\end{document} times a product of constants appearing in Mirzakhani's count of simple closed hyperbolic geodesics. Many of the results in this paper reflect recent discoveries of Delecroix, Goujard, Zograf, and Zorich, but the approach considered here is very different from theirs. We follow conceptual and geometric methods inspired by Mirzakhani's work.

A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a \begin{document}$ 1 $\end{document}-parameter family of genus-\begin{document}$ 2 $\end{document} homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension \begin{document}$ 1 $\end{document}, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.

We study thermodynamic formalism for topologically transitive partially hyperbolic systems in which the center-stable bundle satisfies a bounded expansion property, and show that every potential function satisfying the Bowen property has a unique equilibrium measure. Our method is to use tools from geometric measure theory to construct a suitable family of reference measures on unstable leaves as a dynamical analogue of Hausdorff measure, and then show that the averaged pushforwards of these measures converge to a measure that has the Gibbs property and is the unique equilibrium measure.

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.