Generic Homeomorphisms Research Articles

Generic homeomorphisms have full metric mean dimension

AbstractWe prove that for $C^0$ -generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$ -generic homeomorphism is equal to the dimension of the manifold. In the case of continuous interval maps, we prove that each level set for the metric mean dimension with respect to the Euclidean distance is $C^0$ -dense in the space of continuous endomorphisms of $[0,1]$ with the uniform topology. Moreover, the maximum value is attained at a $C^0$ -generic subset of continuous interval maps and a dense subset of metrics topologically equivalent to the Euclidean distance.

On the rotation sets of generic homeomorphisms on the torus

AbstractWe study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

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.

Quantitative recurrence for generic homeomorphisms

In this article we study quantitative recurrence for generic measure preserving homeomorphisms on euclidian spaces with respect to Lebesgue measure and compact manifolds with respect to Oxtoby-Ulam measures (i.e., it is nonatomic and positive on each open set). As an application we show that the decay of correlations of generic measure preserving homeomorphisms on compact manifolds is slow.

Dynamical properties of spatial discretizations of a generic homeomorphism

This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works on well-chosen examples (such as Gambaudo and Tresser [Some difficulties generated by small sinks in the numerical study of dynamical systems: two examples. Phys. Lett. A 94(9) (1983), 412–414]) show that the dynamical behaviours of dynamical systems and of their discretizations can be quite different. We are interested in generic homeomorphisms of compact manifolds. So our aim is to tackle the following question: can the dynamical properties of a generic homeomorphism be detected on the spatial discretizations of this homeomorphism? We will prove that the dynamics of a single discretization of a generic conservative homeomorphism does not depend on the homeomorphism itself, but rather on the grid used for the discretization. Therefore, dynamical properties of a given generic conservative homeomorphism cannot be detected using a single discretization. Nevertheless, we will also prove that some dynamical features of a generic conservative homeomorphism (such as the set of the periods of all periodic points) can be read on a sequence of finer and finer discretizations.

Rational polygons as rotation sets of generic homeomorphisms of the two torus

We prove the existence of an open and dense set 𝔇⊂Homeo0(𝕋2) (where Homeo0(𝕋2) denotes the set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in 𝔇 is a rational polygon. We also extend this result to the set of axiom A diffeomorphisms in Homeo0(𝕋2). Further, we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(𝕋2).

Ergodic Theory of Generic Continuous Maps

We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measure --- a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps. To further explore the mysterious behaviour of $C^0$ generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.

Dynamics of topologically generic homeomorphisms

Introduction Attractors and chain recurrence Periodic decompositions and adding machines Semicontinuity and homogeneity Crushing arguments Topological horseshoes Generic homeomorphisms Almost equicontinuity Cantor sets The circle Crushing the chain recurrent set Generic homeomorphisms on manifolds Generic mappings on manifolds Bibliography Index.

Generic homeomorphisms do not have the lipschitz shadowing property

Generic homeomorphisms do not have the lipschitz shadowing property

Properties of attractors of generic homeomorphisms

AbstractWe describe several generic properties of homeomorphisms on compact manifolds. These properties concern the attractors of a homeomorphism, its chain recurrent set, and the periodic points.

Generic homeomorphisms have no smallest attractors

We show that if M is a compact manifold then there is a residual subset £% of the set of homeomorphisms on M with the property that if f £31 then / has no smallest attractor (that is, an attractor with the property that none of its proper subsets is also an attractor). Part of the motivation for this result comes from portions of a recent paper by Lewowicz and Tolosa that deal with properties of smallest attractors of generic homeomorphisms. In this paper we consider the set of homeomorphisms of a compact manifold M with metric d ; this set of homeomorphisms is denoted as Hom(Af) and is given the topology of uniform convergence, which makes Hom(Ai) a Baire space. An attractor for / e Hom(Af ) is a compact, nonempty subset A of M with the following property^ there is an open neighborhood U of A satisfying the two conditions (i) f(U) C U and (ii) n>0/(i/) = A. We will refer to such an open set U as an attractor block that determines A (this is slightly different from the standard definition of an attractor block). Note that M itself is an attractor for any / £ Hom(Af) . The attractor A is a smallest attractor for / if no proper subset of A is also an attractor for / ; in other words, a smallest attractor is one that is minimal in the class of all attractors of / when they are partially ordered by inclusion. The main result of this note is that there is a residual subset & of Hom(Af) with the property that if g e 31 then g has no smallest attractor.

Generic Homeomorphisms Have no Smallest Attractors

We show that if M is a compact manifold then there is a residual subset $\mathcal {R}$ of the set of homeomorphisms on M with the property that if $f \in \mathcal {R}$ then f has no smallest attractor (that is, an attractor with the property that none of its proper subsets is also an attractor). Part of the motivation for this result comes from portions of a recent paper by Lewowicz and Tolosa that deal with properties of smallest attractors of generic homeomorphisms.

Generic Homeomorphisms have the Pseudo-Orbit Tracing Property

Generic Homeomorphisms have the Pseudo-Orbit Tracing Property

Generic homeomorphisms have the pseudo-orbit tracing property

Let M (dim M < 3) be a compact manifold. Then a generic f E Homeo(M) satisfies the following: f has the pseudo-orbit tracing property; f is C0 tolerance stable; and f is not topologically stable.