Entropy of homeomorphisms on unimodal inverse limit spaces

Abstract

We prove that every self-homeomorphism h : K s → K s on the inverse limit space K s of the tent map T s with slope s ∈ ( √ 2, 2] has topological entropy h top (h) = |R| log s, where R ∈ Z is such that h and σ R are isotopic. Conclusions on the possible values of the entropy of homeomorphisms of the inverse limit space of a (renormalizable) quadratic map are drawn as well.2000 Mathematics Subject Classification. 54H20, 37B45, 37E05.

Beyond $0$ and $\infty$: A solution to the Barge Entropy Conjecture

“…As a somewhat surprising by-product of its solution (but in agreement with the results described above), it was shown in [8] that the mapping class group of X is Z. This in turn was used to characterize possible values of the topological entropy of homeomorphisms on these spaces [9]. According to it, the entropy is always a non-negative integer multiple of the logarithm of the slope of the tent map.…”