Generic homeomorphisms with shadowing of one-dimensional continua

In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property

Anosov topológicos en el plano

Sea X un espacio topológico, se dice que f : X → X homeomorfismo, es Anosov topológico si es expansivo topológico y tiene la propiedad del sombreado topológico. En el presente trabajo se introducen estos conceptos, y se prueba como resultado principal que un Anosov topológico que preserva orientación en el plano tiene un punto fijo.Let X be a topological space. A homeomorphism f : X --> X is topologically Anosov if it is topologically expansive and has the topological shadowing property. In this work we introduce this concepts and prove our main theorem, that states that an orientation preserving topologically Anosov homeomorphism acting on the plane has a fxed point

Topologically Anosov plane homeomorphisms

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 $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit), and we show that any basin of attraction (or repulsion) must be unbounded.Comment: 10 page

Topologically anosov plane homeomorphisms

This paper deals with classifying the dynamics of topologically Anosov plane homeomorphisms. We prove that a topologically Anosov homeomorphism f: ℝ2 → ℝ2 is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwan- dering set of f reduces to a fixed point, or if there exists an open, connected, simply connected proper subset U such that U ⊂ Int(f(U)), and such that ∪n ≥ 0fn(U) = R2.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)

Generic homeomorphisms with shadowing of one-dimensional continua

In this article, we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property

Linearization of topologically Anosov homeomorphisms of non compact surfaces of genus zero and finite type

We study the dynamics of {\it 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 $f\colon S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \mathbb{R}^2$ 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 \cite{cgx}

Topologically Anosov plane homeomorphisms

This paper deals with classifying the dynamics of {\it topologically Anosov} plane homeomorphisms. We prove that a topologically Anosov homeomorphism $f\colon\mathbb{R}^2 \to \mathbb{R}^2$ 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 \overline {f(U)} \subset \rom{Int} (U), and such that $$\bigcup\limits_{n\leq 0} f^n (U)= \mathbb{R}^2.$$% In the general case, we prove a structure theorem for the $\alpha$-limits of orbits with empty $\omega$-limit (or the $\omega$-limits of orbits with empty $\alpha$-limit)

Topologically anosov plane homeomorphisms

This paper deals with classifying the dynamics of topologically Anosov plane homeomorphisms. We prove that a topologically Anosov homeomorphism f: ℝ2 → ℝ2 is conjugate to a homothety if it is the time one map of a flow. We also obtain results for the cases when the nonwan- dering set of f reduces to a fixed point, or if there exists an open, connected, simply connected proper subset U such that U ⊂ Int(f(U)), and such that ∪n ≥ 0fn(U) = R2.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)