Transport in an area-preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori. A model retaining a discrete set of cantori approaching a boundary circle gives the Markov chain description of Hanson, Cary and Meiss. The inclusion of cantori surrounding island chains, and islands about islands, etc. gives a Markov tree model with a slower decay rate. The survival probability distribution is shown to decay asymptotically as a power law. The decay exponent agrees reasonably well with the computations of Karney and of Chirikov and Shepelyanski.

Markov tree model of transport in area preserving maps

This chapter presents some of the links between automata theory and symbolic dynamics The emphasis is on two particular points The first one is the interplay between some particular classes of automata, such as local automata and results on embeddings of shifts of finite type The second one is the connection between syntactic semigroups and the classification of sofic shifts up to conjugacy

Symbolic dynamics

This paper studies the behavior of harmonic measured foliations on compact Riemann surfaces. Cascades in the dynamics of such a foliation can occur as its relative periods are varied. We show that in the case of genus 2, the bifurcation locus arising from such a variation is a closed, countable set of R that embeds in ω. Resume Nous etudions le comportement des feuilletages mesures harmoniques sur les surfaces de Riemann compactes. Quands les periodes relatives varient, on peut observer des cascades dans la dynamique d’un tel feuilletage. Dans le cas du genre 2, on montre que le lieu de bifurcation resultant d’une telle variation est un sous-ensemble denombrable et ferme de R, qui se plonge dans ω.

/pdf/cascades-in-the-dynamics-of-measured-foliations-2hrmia23af.pdf

Cascades in the Dynamics of Measured Foliations

We consider the restriction of interval exchange transformations to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically, and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable interval exchange maps with zero and non-zero drift vector, and carry out some investigations of their properties. In particular, we look for evidence of the finite decomposition property: each lattice is the union of finitely many orbits.

Geometric representation of interval exchange maps over algebraic number fields

In this paper, we give a necessary condition for an infinite word defined by a non-degenerate interval exchange on three intervals (3iet word) to be invariant by a substitution: a natural parameter associated with this word must be a Sturm number. We deduce some algebraic consequences from this condition concerning the incidence matrix of the associated substitution. As a by-product of our proof, we give a combinatorial characterization of 3iet words.

Sturm numbers and substitution invariance of 3iet words.

This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vectors and carry out some investigations of their properties. In particular we look for evidence of the finite decomposition property on a family of IETs extending the example studied in Lowenstein et al (2007 Dyn. Syst. 22 73–106).

This paper deals with piecewise isometries and their scaling properties. We aim to present a general framework in which piecewise isometric dynamical systems can be fully or partially renormalized. We precisely describe the dynamics on fractal invariant sets by means of a conjugacy with substitution dynamical systems and Bratteli diagrams. We describe their geometrical properties in terms of graph-directed constructions and we compute their Hausdorff dimension. We also present an example of application to a continuous family of piecewise rotations.

Self-similarity in piecewise isometric systems

We study a four-fold symmetric kicked-oscillator map with sawtooth kick function For the values of the kick amplitude λ =2 cos (2π p/q) with rational p / q, the dynamics is known to be pseudochaotic, with no stochastic web of non-zero Lebesgue measure We show that this system can be represented as a piecewise affine map of the unit square—the so-called local map—driving a lattice map We develop a framework for the study of long-time behaviour of the orbits, in the case in which the local map features exact scaling We apply this method to several quadratic irrational values of λ, for which the local map possesses a full Legesgue measure of periodic orbits; these are promoted to either periodic orbits or accelerator modes of the kicked-oscillator map By constrast, the aperiodic orbits of the local map can generate various asymptotic behaviours For some parameter values the orbits remain bounded, while others have excursions which grow logarithmically or as a power of the time In the power-law case, w

/pdf/sticky-orbits-in-a-kicked-oscillator-model-1ncn30itv1.pdf

Sticky orbits in a kicked-oscillator model

We apply methods developed for two-dimensional piecewise isometries to the study of renormalizable interval exchange transformations over an algebraic number field , which lead to dynamics on lattices. We consider the -module generated by the translations of the map. On it, we define an infinite family of discrete vector fields, representing the action of the map over the cosets , which together form an invariant partition of the field . We define a recursive symbolic dynamics, with the property that the eventually periodic sequences coincide with the field elements. We apply this approach to the study of a model introduced by Arnoux and Yoccoz, for which λ−1 is a cubic Pisot number. We show that all cosets of decompose in a highly non-trivial manner into the union of finitely many orbits.

Interval exchange transformations over algebraic number fields: the cubic Arnoux–Yoccoz model

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Papers

Geometric representation of interval exchange maps over algebraic number fields

Geometric representation of interval exchange maps over algebraic number fields

Self-similarity in piecewise isometric systems

Sticky orbits in a kicked-oscillator model

Interval exchange transformations over algebraic number fields: the cubic Arnoux–Yoccoz model