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Gibbs states for non-irreducible countable Markov shifts

Andrei E. Ghenciu, Mario Roy (2013)

Fundamenta Mathematicae

We study Markov shifts over countable (finite or countably infinite) alphabets, i.e. shifts generated by incidence matrices. In particular, we derive necessary and sufficient conditions for the existence of a Gibbs state for a certain class of infinite Markov shifts. We further establish a characterization of the existence, uniqueness and ergodicity of invariant Gibbs states for this class of shifts. Our results generalize the well-known results for finitely irreducible Markov shifts.

Gibbs-Markov-Young structures*, **, ***

Carla L. Dias (2012)

ESAIM: Proceedings

We discuss the geometric structures defined by Young in [9, 10], which are used to prove the existence of an ergodic absolutely continuous invariant probability measure and to study the decay of correlations in expanding or hyperbolic systems on large parts.

Global attractivity of the equilibrium of a nonlinear difference equation

John R. Graef, C. Qian (2002)

Czechoslovak Mathematical Journal

The authors consider the nonlinear difference equation x n + 1 = α x n + x n - k f ( x n - k ) , n = 0 , 1 , ⋯ . 1 where α ∈ ( 0 , 1 ) , k ∈ { 0 , 1 , ⋯ } and f ∈ C 1 [ [ 0 , ∞ ) , [ 0 , ∞ ) ] ( 0 ) with f ' ( x ) < 0 . They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.

Global attractor of a differentiable autonomous system on the plane

Nguyen Van Chau (1995)

Annales Polonici Mathematici

We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.

Global attractors for problems with monotone operators

Alexandre N. Carvalho, Jan W. Cholewa, Tomasz Dlotko (1999)

Bollettino dell'Unione Matematica Italiana

L'esistenza di attrattori globali per equazioni paraboliche semilineari è stata estensivamente studiata da molti autori mentre il caso quasilineare è stato meno considerato e ancora esistono molti problemi aperti. L'obiettivo di questo lavoro è di studiare, da un punto di vista astratto, l'esistenza di attrattori globali per equazioni paraboliche quasilineari con parte principale monotona. I risultati ottenuti vengono applicati a problemi parabolici degeneri del secondo ordine e di ordine superiore....

Global bifurcation of homoclinic trajectories of discrete dynamical systems

Jacobo Pejsachowicz, Robert Skiba (2012)

Open Mathematics

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving topological properties of the asymptotic stable bundles.

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