ISSN:

1930-5311

eISSN:

1930-532X

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## Journal of Modern Dynamics

2020 , Volume 16

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**Abstract:**

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.

*+*[Abstract](373)

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**Abstract:**

We study *joining rigidity* in the class of von Neumann flows with one singularity. They are given by a smooth vector field *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 [

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**Abstract:**

We study the near action of the group

We prove that every finitely generated abelian subgroup of

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.

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**Abstract:**

We show that the number of square-tiled surfaces of genus

*+*[Abstract](252)

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**Abstract:**

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

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**Abstract:**

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.

*+*[Abstract](70)

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**Abstract:**

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.

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