A-coupled-expanding and distributional chaos

The concept of A-coupled-expanding map, which is one of the more natural and useful ideas generalized the horseshoe map, is well known as a criterion of chaos. It is well known that distributional chaos is one of the concepts which reflect strong chaotic behaviour. In this paper, we focus the relations between A-coupled-expanding and distributional chaos. We prove two theorems that give sufficient conditions for a strictly A-coupled-expanding map to be distributionally chaotic in the senses of two kinds, where A is an irreducible transition matrix.Comment: 10 page

Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps

We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these dynamics defined by entropy conditions rather completely, obtaining a complete classification with respect to large entropy measures and a description of their measures with maximum entropy and periodic orbits. These results can in particular be applied to entropy-expanding maps like (x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the meromorphy of the Artin-Mazur zeta function on a large disk. This follows from a similar new result about strongly positively recurrent Markov shifts where the radius of meromorphy is lower bounded by an "entropy at infinity" of the graph.Comment: accepted by Annales de l'Institut Fourier, final revised versio

Discrete Dubrovin Equations and Separation of Variables for Discrete Systems

A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions in terms of Abelian functions of Kleinian type (i.e. the higher-genus analogues of the Weierstrass elliptic functions) is discussed as well as the connections with the method of separation of variables.Comment: Talk presented at the Intl. Conf. on ``Integrability and Chaos in Discrete Systems'', July 2-6, 1997, to appear in: Chaos, Solitons and Fractals, ed. F. Lambert, (Pergamon Press

Rare events, escape rates and quasistationarity: some exact formulae

We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given

Random Wandering Around Homoclinic-like Manifolds in Symplectic Map Chain

We present a method to construct a symplecticity preserving renormalization group map of a chain of weakly nonlinear symplectic maps and obtain a general reduced symplectic map describing its long-time behaviour. It is found that the modulational instability in the reduced map triggers random wandering of orbits around some homoclinic-like manifolds, which is understood as the Bernoulli shifts.Comment: submitted to Prog. Theor. Phy