Exponents, attractors and Hopf decompositions for interval maps. (English) Zbl 0715.58020

Keller, Gerhard

Ergodic Theory Dyn. Syst. 10, No. 4, 717-744 (1990).

Summary: Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:
(1) There is a set \(C_ T\) such that the \(\omega\)-limit of Lebesgue-a.e. point equals \(C_ T\). \(C_ T\) is a finite union of closed intervals or it coincides with the closure of the critical orbit.
(2) There is a constant \(\lambda_ T\) such that \(\lambda_ T=\limsup_{n\to \infty}(1/n)\log | (T^ n)'(x)|\) for Lebesgue- a.e. x.
(3) \(\lambda_ T>0\) if and only if T has an absolutely continuous invariant measure of positive entropy.
(4) \(\lambda_ T\geq \inf \{p^{-1}\log | (T^ p)'(z)|:\) \(T^ pz=z\}\), i.e. uniform hyperbolicity on periodic points implies \(\lambda_ T>0\), and \(\lambda_ T<0\) implies the existence of a stable periodic orbit.

MSC:

37A99	Ergodic theory
28D20	Entropy and other invariants

Keywords:

Hopf-decomposition; regular Markov systems; unimodal interval maps; Schwarzian derivative

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