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 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.