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Dynamics of homeomorphisms on surfaces of genus greater than one. (arXiv:1804.04505v1 [math.DS])

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 manifol查看全文

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