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Lyapunov exponents, KS-entropy and correlation decay in skew product extensions of Bernoulli endomorphisms

S. Siboni (1998)

Bollettino dell'Unione Matematica Italiana

Viene considerata una classe di sistemi dinamici del toro bidimensionale T 2 . Tali sistemi presentano la forma di un prodotto skew fra l'endomorfismo Bernoulli B p ⁢ x = mod ⁡ p ⁢ x , 1 , p ∈ Z ∖ - 1 , 0 , 1 , definito sul toro undidimensionale T 1 ≡ 0 , 1 ed una traslazione del toro stesso. Si dimostra che gli esponenti di Liapunov e l'entropia di Kolmogorov-Sinai della misura di Haar invariante possono essere calcolati esplicitamente. Viene infine discusso il decadimento delle correlazioni per i caratteri.

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

Mark O. Gluzman, Nataliia V. Gorban, Pavlo O. Kasyanov (2015)

Nonautonomous Dynamical Systems

In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction...

Lyapunov quasi-stable trajectories

Changming Ding (2013)

Fundamenta Mathematicae

We introduce the notions of Lyapunov quasi-stability and Zhukovskiĭ quasi-stability of a trajectory in an impulsive semidynamical system defined in a metric space, which are counterparts of corresponding stabilities in the theory of dynamical systems. We initiate the study of fundamental properties of those quasi-stable trajectories, in particular, the structures of their positive limit sets. In fact, we prove that if a trajectory is asymptotically Lyapunov quasi-stable, then its limit set consists...

Maličky-Riečan's entropy as a version of operator entropy

Bartosz Frej (2006)

Fundamenta Mathematicae

The paper deals with the notion of entropy for doubly stochastic operators. It is shown that the entropy defined by Maličky and Riečan in [MR] is equal to the operator entropy proposed in [DF]. Moreover, some continuity properties of the [MR] entropy are established.

Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique

Jean-Pierre Conze, Yves Guivarc'h (2000)

Colloquium Mathematicae

The invariant measures for a Markovian operator corresponding to a random walk, in a random stationary one-dimensional environment defined by a dynamical system, are quasi-invariant measures for the system. We discuss the construction of such measures in the general case and show unicity, under some assumptions, for a rotation on the circle.

Markov partitions for fibre expanding systems

Manfred Denker, Hajo Holzmann (2008)

Colloquium Mathematicae

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

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