Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies
Yongluo Cao Stefano Luzzatto Isabel Rios
We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.
keywords: invariant measures. non-hyperbolic systems Horseshoes Lyapunov exponents
Pressures for asymptotically sub-additive potentials under a mistake function
Wen-Chiao Cheng Yun Zhao Yongluo Cao
This paper defines the pressure for asymptotically sub-additive potentials under a mistake function, including the measure-theoretical and the topological versions. Using the advanced techniques of ergodic theory and topological dynamics, we reveal a variational principle for the new defined topological pressure without any additional conditions on the potentials and the compact metric space.
keywords: asymptotically sub-additive. topological pressure Variational principle
The thermodynamic formalism for sub-additive potentials
Yongluo Cao De-Jun Feng Wen Huang
The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.
keywords: variational principle entropy topological pressure products of matrices. Thermodynamical formalism Lyapunov exponents
Dimension estimates in non-conformal setting
Juan Wang Yongluo Cao Yun Zhao
Given an arbitrary subset of a non-conformal repeller of a $C^1$ map. Using directly the definitions of Hausdorff dimension and Box dimension and of pressure, this paper first proves that the zeros of the topological pressure on this set give its dimension estimates. And without using the estimate of pointwise dimension of an ergodic measure on a non-conformal repeller, it is showed that the zeros of non-additive measure-theoretic pressure give the lower and upper bound of dimension estimate for it. Some results from [22,41] are extended for $C^1$ maps or arbitrary subsets of a non-conformal repeller. And a remark on Rugh's result [34] is also given in this paper.
keywords: Dimension theory ergodic measure measure-theoretic pressure. topological pressure
Continuity of spectral radius over hyperbolic systems
Rui Zou Yongluo Cao Gang Liao

The continuity of joint and generalized spectral radius is proved for Hölder continuous cocycles over hyperbolic systems. We also prove the periodic approximation of Lyapunov exponents for non-invertible non-uniformly hyperbolic systems, and establish the Berger-Wang formula for general dynamical systems.

keywords: Joint spectral radius generalized spectral radius Lyapunov exponents Berger-Wang formula hyperbolic systems

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