CANONICAL REPRESENTATIVES FOR PATTERNS OF TREE MAPS

We define a notion of pattern for finite invariant sets of continuous maps of finite trees. A pattern is essentially a homotopy class relati ve to the finite invariant set. Given such a pattern, we prove that th e class of tree maps which exhibit this pattern admits a canonical rep resentative, that is a tree and a continuous map on this tree, which s atisfies several minimality properties. For instance, it minimizes top ological entropy in its class and its dynamics are minimal in a sense to be defined. We also give a formula to compute the minimal topologic al entropy directly from the combinatorial data of the pattern. Finall y we prove a characterization theorem for zero entropy patterns. (C) 1 997 Elsevier Science Ltd.