Dynamics of homeomorphisms on surfaces of genus greater than one

Abstract

In this work we analyze homeomorphisms and diffeomorphisms homotopic to the identity on a closed surface $S$ of genus greater than one. We present the definition of a homeomorphism $f$ with a system of curves $\mathscr{C}$ and show that for such homeomorphism, the natural lift $\tilde{f}$ of $f$ to the universal cover of $S$ has complicated and rich dynamics. In this context we generalize results that hold for homeomorphisms of the torus with rotation set with non-empty interior. In particular, we prove that when $f$ is a homeomorphism with a system of curves $\mathscr{C}$, then for every deck transformation on the universal cover of $S$, there exists a periodic point in $S$ associated to it. In the case that $f$ is a $C^{1+\epsilon}$-diffeomorphism with a system of curves $\mathscr{C}$, there exists a hyperbolic $\tilde{f}$-periodic saddle point in the universal cover of $S$ such that the unstable manifold of this point has a topological transverse intersection with the stable manifold of translations of this point under all deck transformations. We also show that the homological rotation set of a $C^{1+\epsilon}$-diffeomorphism with a system of curves $\mathscr{C}$ is a compact convex set. Moreover, all points in its interior are realized by compact $f$-invariant sets, periodic points in the rational case.