Distortion bounds for $C^{2+η}$ unimodal maps
| dc.creator | Todd, Mike | |
| dc.date | 2006-01-23 | |
| dc.date | 2006-08-08 | |
| dc.date.accessioned | 2026-07-07T07:45:02Z | |
| dc.date.available | 2026-07-07T07:45:02Z | |
| dc.description | We obtain estimates for derivative and cross--ratio distortion for $C^{2+η}$ (any $η>0$) unimodal maps with non--flat critical points. We do not require any `Schwarzian--like' condition. For two intervals $J \subset T$, the cross--ratio is defined as the value $$B(T,J):=\frac{|T||J|}{|L||R|}$$ where $L,R$ are the left and right connected components of $T\setminus J$ respectively. For an interval map $g$ \st $g_T:T \to \RRR$ is a diffeomorphism, we consider the cross--ratio distortion to be $$B(g,T, J):=\frac{B(g(T),g(J))}{B(T,J)}.$$ We prove that for all $0<K<1$ there exists some interval $I_0$ around the critical point \st for any intervals $J \subset T$, if $f^n|_T$ is a diffeomorphism and $f^n(T) \subset I_0$ then $$B(f^n, T, J)> K.$$ Then the distortion of derivatives of $f^n|_J$ can be estimated with the Koebe Lemma in terms of $K$ and $B(f^n(T),f^n(J))$. This tool is commonly used to study topological, geometric and ergodic properties of $f$. This extends a result of Kozlovski. | |
| dc.description | 5 figures. Typos corrected, and some clarifications made; principally to the first part of Section 4. To appear in Fundamenta Mathematicae | |
| dc.identifier | https://arxiv.org/abs/math/0601564 | |
| dc.identifier | http://arxiv.org/abs/math/0601564 | |
| dc.identifier | Fund Math 193 (2007) 37-77 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/123407 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 37E05 | |
| dc.title | Distortion bounds for $C^{2+η}$ unimodal maps | |
| dc.type | text |