# Genericity of historic behavior for maps and flows

@article{Carvalho2021GenericityOH,
title={Genericity of historic behavior for maps and flows},
author={Maria Carvalho and Paulo Varandas},
journal={Nonlinearity},
year={2021},
volume={34},
pages={7030 - 7044}
}
• Published 2 July 2021
• Mathematics
• Nonlinearity
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…
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## References

SHOWING 1-10 OF 68 REFERENCES

### Nonuniform hyperbolicity for C1-generic diffeomorphisms

• Mathematics
• 2011
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

• Mathematics
• 2017
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

• Mathematics
• 2008
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

• Mathematics
Bulletin of the Brazilian Mathematical Society, New Series
• 2019
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

• Mathematics
• 2000
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

### Partial hyperbolicity and specification

• Mathematics
• 2013
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