Getting rid of the negative Schwarzian derivative condition
By Oleg Kozlovski. 2000
76 citations recorded by the source corpus.
In this paper we will show that the assumption on the negative Schwarzian derivative is redundant in the case of C 3 unimodal maps with a non∞at critical point. The following theorem will be proved: For any C 3 unimodal map of an interval with a non∞at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative Schwarzian derivative. Another theorem proved in the paper provides useful cross-ratio estimates. Thus, all theorems proved only for unimodal maps with negative Schwarzian derivative can be easily generalized.
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a587125f32e7201d73eb9d6a9bc7b989fe687bca
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In Geometry and Topology
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Real bounds, ergodicity and negative Schwarzian for multimodal maps (2004, unknown): “The last part of Theorem C generalizes the results of Kozlovski [6] (which he proved for unimodal maps) to general multimodal maps.”
- Metric attractors for smooth unimodal maps (2004, background): “The main ingredient is the decay of geometry.”
- Periodic attractors of perturbed one-dimensional maps (2013, unknown): “2 literally follows the proof in [Koz00] or [VV04].”
- Equilibrium measures for maps with inducing schemes (2006, background): “The negative Schwarzian derivative assumption is not necessary to prove distortion bounds for C 3-unimodal maps with no neutral periodic cycles [21] (and even C 2+η unimodal maps, see [40]), or for C 3 multimodal maps [42].”