10 problems

Let $X$ be a Venice mask supported on a compact $3$-manifold $M$, and let $H_1$ and $H_2$ be two different homoclinic classes in $M(X)$. Set $\Lambda=H_1\cap H_2$. Write…

Let $X$ be a Venice mask supported on a compact $3$-manifold $M$. Let $H_1$ and $H_2$ be two different homoclinic classes in $M(X)$), let $\Lambda=H_1\cap H_2$, and suppose that th…

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 $X$ 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 $M$. For a periodic point $p$, its homoclinic cla…

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…

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…