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### Open Access Journals

DCDS

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.
DCDS

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.

DCDS

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.

DCDS

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.

DCDS

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.

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