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Intrinsic ergodicity of smooth interval maps. (English) Zbl 0889.28009
Let \(f\) be a \(C^\infty\) self-map of the interval \([0,1]\) that may have infinitely many critical points, i.e., points where the derivative changes its sign. The main result of this paper is the following Theorem: If \(f\) has strictly positive topological entropy, then the set of ergodic measures of maximal entropy for \(f\) is finite and non-empty, and each topologically transitive component of \(f\) carries at most one such measure. This result generalizes work of Hofbauer (who proved the same result for maps with finitely many critical points but without smoothness assumptions). It makes essential use of the topological spectral decomposition for continuous \(f\) by Blokh, of bounds on the upper semicontinuity defect of the entropy in terms of local entropy by Newhouse and by Yomdin, of a version of Hofbauer’s Markov diagrams adapted to maps with infinitely many critical points, and of Gurevic’s results on measures of maximal entropy for countable state topological Markov chains.
In an appendix, the author constructs examples of \(C^r\) interval maps showing that the \(C^\infty\) assumption made on \(f\) is indispensible for both the existence as well as the finiteness result for the set of ergodic measures of maximal entropy for \(f\).

28D05 Measure-preserving transformations
37A99 Ergodic theory
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
28D15 General groups of measure-preserving transformations
Full Text: DOI
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