# Problem 91

## Primary tabs

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

## Tags

Tags:

### FL

What does this mean?

### On the set of continuous

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].

### References

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.  Israel J. Math.. 57:285--300.