Distortion Bounds for C Unimodal Maps
نویسنده
چکیده
We obtain estimates for derivative and cross–ratio distortion for C (any η > 0) unimodal maps with non–flat critical points. We do not require any ‘Schwarzian–like’ condition. For two intervals J ⊂ T , the cross–ratio is defined as the value B(T, J) := |T ||J | |L||R| where L, R are the left and right connected components of T \J respectively. For an interval map g such that gT : T → R is a diffeomorphism, we consider the cross–ratio distortion to be B(g, T, J) := B(g(T ), g(J)) B(T, J) . We prove that for all 0 < K < 1 there exists some interval I0 around the critical point such that for any intervals J ⊂ T , if f|T is a diffeomorphism and f(T ) ⊂ I0 then B(f, T, J) > K. Then the distortion of derivatives of f|J can be estimated with the Koebe Lemma in terms of K and B(f(T ), f(J)). This tool is commonly used to study topological, geometric and ergodic properties of f . This extends a result of Kozlovski.
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