The coupling of dynamics in couplied map lattices.
The dynamics of complex polynomials and automorphisms of the shift.
The existence of an a.c.i.p.m. for an expanding map of the interval; the study of a counterexample
The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards.
The forcing relation for horseshoe braid types.
The Lipschitz condition for the conjugacies of Feigenbaum-like mappings
The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift.
The metric space of geodesic laminations on a surface. I.
The minimum tree for a given zero-entropy period.
The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century
The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.
The real 3x+1 problem
The recurrence dimension for piecewise monotonic maps of the interval
We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval , giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure...
The Ruelle rotation of Killing vector fields
We present an explicit formula for the Ruelle rotation of a nonsingular Killing vector field of a closed, oriented, Riemannian 3-manifold, with respect to Riemannian volume.
The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos
We discuss the existence of an uncountable strongly chaotic set of a continuous self-map on a compact metric space. It is proved that if a continuous self-map on a compact metric space has a regular shift invariant set then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.
The structure of disjoint iteration groups on the circle
The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle , that is, families of homeomorphisms such that and each either is the identity mapping or has no fixed point ( is an arbitrary -divisible nontrivial (i.e., ) abelian group).
The topological entropy versus level sets for interval maps
We answer affirmatively Coven's question [PC]: Suppose f: I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2?
The topological entropy versus level sets for interval maps (part II)
Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.
The Williams conjecture is false for irreducible subshifts.
Thickness measures for Cantor sets.
Thurston unbounded earthquake maps.