Expansivity implies existence of Hölder continuous Lyapunov function
Łukasz Struski Jacek Tabor
Discrete & Continuous Dynamical Systems - B 2017, 22(9): 3575-3589 doi: 10.3934/dcdsb.2017180

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.

keywords: Lyapunov function expansive map Hölder continuous cone-fields
Computational hyperbolicity
Marcin Mazur Jacek Tabor
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1175-1189 doi: 10.3934/dcds.2011.29.1175
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.
keywords: computer assisted proof. semihyperbolicity Hyperbolicity
Cone-fields without constant orbit core dimension
Łukasz Struski Jacek Tabor Tomasz Kułaga
Discrete & Continuous Dynamical Systems - A 2012, 32(10): 3651-3664 doi: 10.3934/dcds.2012.32.3651
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.
keywords: Cone-field dominated splitting expansive map shadowing property. hyperbolicity
Semi-hyperbolicity and hyperbolicity
Marcin Mazur Jacek Tabor Piotr Kościelniak
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 1029-1038 doi: 10.3934/dcds.2008.20.1029
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.
keywords: numerical computations. Hyperbolicity semi-hyperbolicity

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