ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 111 – 117

Strictly Ergodic Patterns and Entropy for Interval Maps
J. Bobok

Abstract.  Let $\M$ be the set of all pairs $(T,g)$ such that $T\subset \RR$ is compact, $g: T\to T$ is continuous, $g$ is minimal on $T$ and has a piecewise monotone extension to $\conv T$. Two pairs $(T,g),(S,f)$ from $\M$ are equivalent -- $(T,g)\sim (S,f)$ -- if the map $h:\orb(\min T,g)\to \orb(\min S,f)$ defined for each $m\in \NN_0$ by $h(g^m(\min T))=\linebreak =f^m(\min S)$ is increasing on $\orb(\min T,g)$. An equivalence class of this relation is called a minimal (oriented) pattern. Such a pattern $A\in\M_{\sim}$ is strictly ergodic if for some $(T,g)\in A$ there is exactly one $g$-invariant normalized Borel measure $\mu$ satisfying $\supp\mu=T$. A pattern $A$ is exhibited by a continuous interval map $f:I\to I$ if there is a set $T\subset I$ such that $(T,f|T)=(T,g)\in A$. Using the fact that for two equivalent pairs $(T,g),(S,f)\in A$ their topological entropies $\ent(g,T)$ and $\ent(f,S)$ equal we can define the lower topological entropy $\inent(A)$ of a minimal pattern $A$ as that common value. We show that the topological entropy $\ent(f,I)$ of a continuous interval map $f:I\to I$ is the supremum of lower entropies of strictly ergodic patterns exhibited by $f$.

AMS subject classification:  26A18, 37A05, 37B40, 37E05
Keywords:  Interval map, strictly ergodic pattern, topological entropy