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In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t, and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates:

$$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$

where h is the topological entropy of the geodesic flow. We also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

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Counting closed geodesics on rank-one manifolds without focal points

Measures of maximal entropy of bounded density shifts

Geodesic Flows Modelled by Expansive Flows

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3 results

Counting closed geodesics on rank-one manifolds without focal points

Measures of maximal entropy of bounded density shifts

Geodesic Flows Modelled by Expansive Flows