13 problems

Let $M$ be a compact surface, let $\operatorname{Diff}_\mu^r(M)$ denote the space of $C^r$ area-preserving diffeomorphisms of $M$, and let $W^s(p)$ and $W^u(p)$ be the stable and u…

Let $f$ be a $C^r$ diffeomorphism, with $r>1$, on a compact surface $M$. Write $h_{\rm top}(f)$ for its topological entropy and $\lambda^{+}(f)$ for the relevant positive Lyapunov-…

Effective SPR conjecture. For every $1>\alpha_1>\alpha_2>0$, there exist $\beta_0>0$ and $C>0$ such that, for every ergodic measure $\mu$ of $f$ satisfying

Let $M$ be a compact surface, let $f\colon M\to M$ be a diffeomorphism, let $P_n(f)=\{x\in M:x=f^n(x)\}$, and let $h_{\operatorname{top}}(f)$ denote the topological entropy of $f$.…

For each genus $g$, let $\delta_{g,0}$ be the minimum dilatation of pseudo-Anosov maps on the closed surface of genus $g$. Let $\mu=\frac{1+\sqrt{5}}{2}\approx1.618$ be the golden…

Let $S$ be a compact surface, let $f\in \operatorname{Diff}^{1}(S)$ be a diffeomorphism, and let the potential be Holder continuous. A phase transition is a failure of analyticity…

Let $f$ be a $C^r$ surface diffeomorphism, where $1<r<\infty$, and let $h_{\operatorname{top}}(f)$ be its topological entropy. A measure maximizing entropy is an invariant probabil…

Let $f$ be a $C^r$ diffeomorphism of a closed surface, where $1<r<\infty$, and let $\lambda_{\min}(f)$ denote the minimum expansion rate appearing in the entropy estimates. A measu…

Let $f$ be an orientation-preserving, real-analytic, Lebesgue measure-preserving diffeomorphism of the 2-sphere $\mathbb{S}^2$ with only two periodic points, which are necessarily…

Let $M$ be a real compact surface, and let $k\geq 1$. For $f\in\mathrm{Diff}^k(M)$, a homoclinic tangency means a tangency between the stable and unstable manifolds of some saddle…

Let $f:M\to M$ be a $C^\infty$ surface diffeomorphism, and let $h_{\operatorname{top}}(f)$ denote its topological entropy. A maximum measure is an invariant probability measure who…

Soit $M$ une variété compacte de dimension $2$, et notons $\operatorname{Diff}^1(M)$ l'espace des difféomorphismes de classe $C^1$. Un difféomorphisme est hyperbolique s'il satisfa…

Soit $S$ une surface compacte, et notons $\operatorname{Diff}^1(S)$ l'espace des difféomorphismes de classe $C^1$ de $S$. Un difféomorphisme est hyperbolique s'il satisfait la défi…