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### Open Access Journals

DCDS-B

The Lyapunov function is a very useful tool in the theory of dynamical systems, in particular in the study of the stability of an equilibrium point. In this paper we construct a locally Hölder continuous Lyapunov function for a uniformly expansive set for a map *f* in a metric space *X*. In the construction a basic role is played by the functions defining the stable and unstable cone-fields. As a tool we also use the approximately quasiconvex functions.

DCDS

Using semihyperbolicity as a basic tool, we provide a general computer assisted method for verifying
hyperbolicity of a given set.
As a consequence we obtain that the Hénon attractor is hyperbolic for some parameter values.

DCDS

As is well-known, the existence of a cone-field with constant orbit core dimension is, roughly speaking, equivalent to hyperbolicity, and consequently guarantees expansivity and shadowing.
In this paper we study the case when the given cone-field does not have the constant orbit core dimension. It occurs that we still obtain expansivity even in general metric spaces.

**Main Result.***Let $X$ be a metric space and let $f:X \rightharpoonup X$ be a given partial map. If there exists a uniform cone-field on $X$ such that $f$ is*cone-hyperbolic,*then $f$ is*uniformly expansive,*i.e. there exists $N \in \mathbb{N}$, $\lambda \in [0,1)$ and $\epsilon > 0$ such that for all orbits*$\mathrm{x},\mathrm{v}:{-N,\ldots,N} \to X$ \[ d_{\sup}(\mathrm{x},\mathrm{v}) \leq \epsilon \Longrightarrow d(\mathrm{x}_0,\mathrm{v}_0) \leq \lambda d_{\sup}(\mathrm{x},\mathrm{v}). \] } We also show a simple example of a cone hyperbolic orbit in $\mathbb{R}^3$ which does not have the shadowing property.
DCDS

We prove that for $\mathcal{C}^1$-diffeomorfisms
semi-hyperbolicity of an invariant set implies its
hyperbolicity. Moreover, we provide some exact estimations of
hyperbolicity constants by semi-hyperbolicity ones, which can be
useful in strict numerical computations.

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