Problem 91

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Is the topological entropy continuous on $C^1$ expanding maps of the interval? (a.e. continuous?)



What does this mean?

Dropping the expanding condition (which cannot be satisfied on the interval), on the set of continuous interval maps, the topological entropy is lower-semicontinuous but not upper-semicontinuous [1]. It is continuous among $C^\infty$ interval maps, as a consequence of Yomdin's theory [2].


  1. [misiurewicz2001jumps] [Anonymous].  2001.  Possible jumps of entropy for interval maps. Possible jumps of entropy for interval maps. 2:289--306.
  2. [yomdin1987volume] Yomdin Y..  1987.  Volume growth and entropy. Israel J. Math.. 57:285--300.

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