**
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

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Acta Mathematica Universitatis Comenianae

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