arXiv subjects
Publications and source records attributed to Gonzalo Cousillas.
This paper studies the behavior of dynamical systems in non-compact spaces, specifically focusing on the concepts of global attractors and shadowing. Let $K$ be a compact global attractor. We show that the shadowing property holds in certain types of dynamical systems in non-compact spaces if and only if $K$ is trivial.
We study the dynamics of Topologically Anosov homeomorphisms of non compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f:S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \R ^ 2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in a previous work.
In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.
In this paper we show that every homeomorphism of the plane with the topological shadowing property has a fixed point. Also, we show that a linear isomorphism of an Euclidean space has the topological shadowing property if and only if the origin is an attractor or a repeller.
This paper deals with classifying the dynamics of {\it Topologically Anosov} plane homeomorphisms. We prove that a Topologically Anosov homeomorphism $f:\mathbb{R}^2 \to \mathbb{R}^2$ is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwandering set of $f$ reduces to a fixed point, or if there exists an open, connected, simply connected proper subset $U$ such that $U \subset \mathrm{Int}(\overline {f(U)})$, and such that $ \cup_{n\geq 0} f^n (U)= \mathbb{R}^2$. In the general case, we prove a structure theorem for the $α$-limits of orbits with empty $ω$-limit (or the $ω$-limits of orbits with empty $α$-limit), and we show that any basin of attraction (or repulsion) must be unbounded.