16 problems

Let $f$ be a $C^2$-diffeomorphism and let $K$ be a compact locally maximal invariant set of $f$ admitting a dominated splitting … where $F$ has dimension $1$ and $E^s$ is uniformly…

Let $f$ be a $C^r$ diffeomorphism of a compact surface. For $0<t\le h_{\rm top}(f)$, let $\mathcal H(f,t)$ be the set of homoclinic classes of measures $C$ satisfying $h(f,C)\ge t$…

Let $f$ be a $C^r$ diffeomorphism of a compact surface. For $0<t\le h_{\rm top}(f)$, let $\mathcal H(f,t)$ be the set of homoclinic classes of measures $C$ satisfying $h(f,C)\ge t$…

For a hyperbolic periodic point $p$, its homoclinic class $H(p)$ is the closure of the set of periodic points $q$ homoclinically related to $p$, meaning that…

Let $M$ be a manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $bb$ be a homoclinic class that is away from homoclinic tangencies and contains singularities. A subs…

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

Let $X$ be a Venice mask on a compact manifold, and let $M(X)$ denote its maximal invariant set. A homoclinic class is the homoclinic class associated with a periodic orbit. The fi…

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms of $M$. A heterodimensional cycle is a pair of hyper…

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms of $M$. A non-hyperbolic homoclinic class is a homoc…

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^1(M)$ be the space of $C^1$ diffeomorphisms of $M$. For a periodic point $p$, its homoclinic cla…

Characteristic-property conjecture. Existence of non-hyperbolic ergodic measures is a characteristic property of non-hyperbolic homoclinic classes.

Let $M$ be a compact manifold and let $\operatorname{Diff}^1(M)$ be the space of $C^1$-diffeomorphisms. A residual subset is a countable intersection of open dense subsets, and the…

Generic sink-or-source conjecture. For any integer $r\geq 1$, for any diffeomorphism $f$ in a residual subset $\mathcal R\subset\operatorname{Diff}^r(M)$, a homoclinic class…

Bonatti's conjecture. Any ergodic measure supported on $H(p,f)$ can be approximated in the weak topology by periodic measures supported on $H(p,f)$.

Let $P$ be a hyperbolic saddle of a diffeomorphism $f$. Assume that for every diffeomorphism $g$ that is $C^1$-close to $f$, there is no heterodimensional cycle associated to the c…

Let $M$ be a compact connected boundaryless manifold of dimension $d$, and let $\operatorname{Diff}^{1}(M)$ be the space of diffeomorphisms of $M$ with the $C^1$ topology. A homocl…