Lluís Alsedà

Combinatorial dynamics and entropy in dimension one

Preliminaries: general notation graphs, loops and cycles. Interval maps: the Sharkovskii Theorem maps with the prescribed set of periods forcing relation patterns for interval maps antisymmetry of the forcing relation P-monotone maps and oriented patterns consequences of Theorem 2.6.13 stability of patterns and periods primary patterns extensions characterization of primary oriented patterns more about primary oriented patterns. Circle maps: liftings and degree of circle maps lifted cycles cycles and lifted cycles periods for maps of degree different from -1, 0 and 1 periods for maps of degree 0 periods for maps of degree -1 rotation numbers and twist lifted cycles estimate of a rotation interval periods for maps of degree 1 maps of degree 1 with the prescribed set of periods other results. Appendix: lifted patterns. Entropy: definitions entropy for interval maps horseshoes entropy of cycles continuity properties of the entropy semiconjugacy to a map of a constant slope entropy for circle maps proof of Theorem 4.7.3.

Periodic orbits of maps of Y

We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of Y = (z e C: z3 e [0,1]} into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.

Entropy of transitive tree maps

Abstract We obtain lower bounds for the topological entropy of transitive self-maps of trees, depending on the number of endpoints and on the number of edges of the tree.

Periods for triangular maps

We study the sets of periods of triangular maps on a cartesian product of arbitrary spaces. As a consequence we extend Kloedens Theorem (in a 1979 paper) to a class of triangular maps on cartesian products of intervals and circles. We also show that, in some sense, this is the more general situation in which the Sharkovskiĭ ordering gives the periodic structure of triangular maps.

Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

Let f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p / q ≠ 0. If ( p , q ) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p / q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2π p / q ). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Minimal periodic orbits for continuous maps of the interval

For continuous maps of the interval into itself, Sarkovskiis Theorem gives the notion of minimal periodic orbit. We complete the characterization of the behavior of minimal periodic orbits. Also, we show for unimodal maps that the min-max essentially describes the behavior of minimal periodic orbits.

ON THE STRUCTURE OF THE ω-LIMIT SETS FOR CONTINUOUS MAPS OF THE INTERVAL

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point xα∈I such that its center has depth at least α. This improves a result by [Sharkovskii, 1966].

Canonical representatives for patterns of tree maps

Abstract We define a notion of pattern for finite invariant sets of continuous maps of finite trees. A pattern is essentially a homotopy class relative to the finite invariant set. Given such a pattern, we prove that the class of tree maps which exhibit this pattern admits a canonical representative, that is a tree and a continuous map on this tree, which satisfies several minimality properties. For instance, it minimizes topological entropy in its class and its dynamics are minimal in a sense to be defined. We also give a formula to compute the minimal topological entropy directly from the combinatorial data of the pattern. Finally we prove a characterization theorem for zero entropy patterns.

Lower bounds of the topological entropy for continuous maps of the circle of degree one

The authors give the best lower bound of the topological entropy of a continuous map of the circle of degree one, as a function of the rotation interval. Also, they obtain as a corollary the theorem of Ito (1982) which gives the best lower bound of the topological entropy depending on the set of periods.

LOW-DIMENSIONAL COMBINATORIAL DYNAMICS

The aim of this paper is to give an account of some of the progress made in these last years in the combinatorial low-dimensional dynamics and to suggest some research directions and open problems.