Pere Mumbrú

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.

Periodic behavior on trees

We characterize the set of periods for tree maps. More precisely, we prove that the set of periods of any tree map

Sets of Periods for Piecewise Monotone Tree Maps

f:T \to T

Minimizing topological entropy for continuous maps on graphs

is the union of finitely many initial segments of Baldwins orderings

Patterns and minimal dynamics for graph maps

_p{\geq}

A note on the periodic orbits and topological entropy of graph maps

and a finite set

On the minimal models for graph maps

\mathcal{F}

Stable heteroclinic cycles and symbolic dynamics

. The possible values of p and explicit upper bounds for the size of

On sizes and shapes of lattice figures

\mathcal{F}

Minimal dynamics for tree maps

are given in terms of the combinatorial properties of the tree T . Conversely, given any set