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
[misiurewicz2001jumps]
.
It is continuous among $C^\infty$ interval maps, as a consequence of Yomdin’s
theory
[yomdin1987volume]
.
Comments
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 [misiurewicz2001jumps] . It is continuous among $C^\infty$ interval maps, as a consequence of Yomdin’s theory [yomdin1987volume] .