See Buzzi, Jérôme. Intrinsic ergodicity of smooth interval maps. Israel J.
Math. 100 (1997), 125–161. And also Yomdin, Y. Volume growth and entropy.
Israel J. Math. 57 (1987), no. 3, 285–300.
jeromebuzzi (2017-08-01 23:40:37 -0700 -0700):
M. Misiurewicz
[misiurewicz1973NoMME]
built $C^r$ diffeomorphisms
without measures maximizing the entropy. These were the first known examples
of non entropy-expansive maps. Nowadays, it is easy to construct a real-
analytic example, e.g., the map $f:[-1,1]\to[-1,1]$, $x\mapsto 1-2x^2$.
Indeed, for any $\epsilon>0$, the “entropy below scale $\epsilon$” is
positive. This can be shown by considering an orbit visiting
$(-\epsilon/2,-\epsilon/4)\cup(\epsilon/4,\epsilon/2)$ with positive
frequency. On the other hand, the weaker property of asymptotic entropy-
expansiveness, i.e., that the “entropy below scale $\epsilon$” converges to
zero as $\epsilon$ goes to $0$, holds for every $C^\infty$-map of a compact
manifold and in particular for any real-analytic map
[buzzi1997intrinsic]
. For more precise results, see
[yomdin1991analytic]
and
[burguet2015entropyRate]
.
Comments
See Buzzi, Jérôme. Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1997), 125–161. And also Yomdin, Y. Volume growth and entropy. Israel J. Math. 57 (1987), no. 3, 285–300.
M. Misiurewicz [misiurewicz1973NoMME] built $C^r$ diffeomorphisms without measures maximizing the entropy. These were the first known examples of non entropy-expansive maps. Nowadays, it is easy to construct a real- analytic example, e.g., the map $f:[-1,1]\to[-1,1]$, $x\mapsto 1-2x^2$. Indeed, for any $\epsilon>0$, the “entropy below scale $\epsilon$” is positive. This can be shown by considering an orbit visiting $(-\epsilon/2,-\epsilon/4)\cup(\epsilon/4,\epsilon/2)$ with positive frequency. On the other hand, the weaker property of asymptotic entropy- expansiveness, i.e., that the “entropy below scale $\epsilon$” converges to zero as $\epsilon$ goes to $0$, holds for every $C^\infty$-map of a compact manifold and in particular for any real-analytic map [buzzi1997intrinsic] . For more precise results, see [yomdin1991analytic] and [burguet2015entropyRate] .