Abstract:

In this talk we will consider tree maps (a "tree map" is a continuous map from a tree into itself). On the other hand, we will look at the (topological) entropy of such maps. The "entropy" $h(f)$ of a map $f$ is a well-known topological invariant which measures the dynamical complexity of $f.$ It is a nonnegative real number (or infinity). Zero-entropy maps are dynamically trivial, while positive-entropy maps are chaotic in the sense of Li and Yorke. It is well known that any interval map (an interval is the only tree with $2$ endpoints) having a periodic orbit of period $3$ has positive entropy. On the other hand, it is easy to define a zero-entropy map on a $3$-star (the only tree with $3$ endpoints) having a periodic orbit of period $3:$ take, for instance, a rigid rotation fixing the branching point. So one may wonder, for any natural number $n,$ which is the minimum number of endpoints $e(n)$ of a tree admitting a zero-entropy map with a periodic orbit of period $n?$ We answer this question by giving a general formula for $e(n).$ As a corollary, we get a criterion to decide whether a map $f$ defined on a tree with $e$ endpoints has positive entropy: if $f$ has a periodic orbit of period $m$ with $e(m) > e,$ then the entropy of $f$ is positive.