26 problems

Buzzi's conjecture. The diffeomorphism $f$ has at most countably many ergodic measures of maximal entropy.

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 $f$.…

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 $f$ 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

Palis's conjecture. The space of two-dimensional $C^r$ diffeomorphisms with $r\ge 2$ is the closure of the union of just two open sets: axiom A systems and the Newhouse domain. Thi…

Let $f$ be a $\mathcal{C}^r$ diffeomorphism and let $(\mu_n)_{n\geq1}$ be any sequence of saddle ergodic invariant measures such that $\mu_i$ is not homoclinically related to…

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 $\Sigma$ be a surface, and let $f\colon \Sigma\to\Sigma$ be a tame self-diffeomorphism, meaning that for every pair of homotopy classes of closed curves $\alpha$ and $\beta$ in…

A measurable pseudo-Anosov map is the type of dynamical system described in the source. A 0-entropy quotient is the quotient associated to a surface diffeomorphism; a homoclinic cl…

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 $S$ be a surface and let $\varphi\colon S\to S$ be pseudo-Anosov. Let $[r]\in\mathcal X_{\mathrm{SL}_2(\mathbb C)}(S)$ be a $\varphi$-invariant character lying, in the fixed-po…

Consider differentiable surface systems that do not preserve a volume form. Smale's conjecture. An open and dense set of such systems should be uniformly hyperbolic. This scenario…

Let $f$ be a $C^r$ surface diffeomorphism, where $1<r<\infty$, let $h_{\operatorname{top}}(f)$ denote its topological entropy, and let…

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…

A real analytic pseudo-rotation is a real analytic area-preserving diffeomorphism of the disk having the pseudo-rotation properties described in the source. The paper notes that no…

Entropy bound conjecture. The Main Theorem should also hold for surface diffeomorphisms, namely

Palis's finite-attractors conjecture. With total probability in the parameter line, the corresponding diffeomorphisms do not display infinitely many attractors, in particular sinks…

Le Roux's conjecture. The point $z_0$ is accumulated by periodic orbits.

Let $f$ be a $C^\infty$ surface diffeomorphism with positive topological entropy, and consider a measure of maximal entropy for $f$. Buzzi's conjecture. The measure of maximal entr…

Let $M$ be a compact orientable smooth surface and let $f:M\to M$ be a $C^\infty$ diffeomorphism. An ergodic invariant measure of maximal entropy is an ergodic invariant probabilit…

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…

Let $M$ be a compact surface, and let $\operatorname{Diff}^1(M)$ denote the space of $C^1$ diffeomorphisms of $M$. An Axiom A surface diffeomorphism is a surface diffeomorphism sat…

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…