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Invariant measures exist under a summability condition for unimodal maps. (English) Zbl 0736.58030
Let \(f\) be a universal \(C^ 3\)-function on the interval with negative Schwarzian derivative. If at the critical value \(c\) the iterates \(f^ n=f\circ\ldots\circ f\) of \(f\) have the property that \((f^ n)'(f(c))\) tends to infinity “fast enough” then \(f\) admits a unique absolutely continuous (w.r.t. the Lebesgue measure) invariant probability measure. This generalizes a result of Collet and Eckmann.

MSC:
37A99 Ergodic theory
37D99 Dynamical systems with hyperbolic behavior
28D20 Entropy and other invariants
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