Genericity of historic behavior for maps and flows

  title={Genericity of historic behavior for maps and flows},
  author={Maria Carvalho and Paulo Varandas},
  pages={7030 - 7044}
We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose… 

Sensitivity and historic behavior for continuous maps on Baire metric spaces

We introduce a notion of sensitivity, with respect to a continuous bounded observable, which provides a sufficient condition for a continuous map, acting on a Baire metric space, to exhibit a Baire

Ergodic Formalism for topological Attractors and historic behavior

. We introduce the concept of Baire Ergodicity and Ergodic Formalism. We use them to study topological and statistical attractors, in particular to establish the existence and finiteness of such

On multifractal analysis and large deviations of singular-hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class

Entropy of irregular points that are not uniformly hyperbolic

In this article we prove that for a C diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy

Ergodic Average Of Typical Orbits And Typical Functions

In this article we mainly aim to know what kind of asymptotic behavior of typical orbits can display. For example, we show in any transitive system, the emprical measures of a typical orbit can cover

Stable and Historic Behavior in Replicator Equations Generated by Similar-Order Preserving Mappings

A BSTRACT . One could observe drastically different dynamics of zero-sum and non-zero- sum games under replicator equations . In zero-sum games, heteroclinic cycles naturally occur whenever the


We consider the set of points with historic behavior (which is also called the irregular set) for continuous flows and suspension flows. In this paper under the hypothesis that (Xt)t is a continuous



Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with

Introduction to Ergodic theory

Hyperbolic dynamics studies the iteration of maps on sets with some type of Lipschitz structure used to measure distance. In a hyperbolic system, some directions are uniformly contracted and others

Topological entropy of level sets of empirical measures for non-uniformly expanding maps

In this article we obtain a variational principle for saturated sets for maps with some non-uniform specification properties. More precisely, we prove that the topological entropy of saturated sets

Singular-hyperbolic attractors are chaotic

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with

Nonexistence of Lyapunov exponents for matrix cocycles

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect

Abundance of Wild Historic Behavior

Using Caratheodory measures, we associate to each positive orbit of a measurable map f , a Borel measure $$\eta _{x}$$ η x . We show that $$\eta _{x}$$ η x is f -invariant whenever f is continuous or

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages

Gluing orbit property and partial hyperbolicity

Partial hyperbolicity and specification

We study the specification property for partially hyperbolic dy- namical systems. In particular, we show that if a partially hyperbolic diffeo- morphism has two saddles with different indices, and