Planar embeddings of inverse limit spaces of unimodal maps
A dynamical system \(f:X\to X\) can be represented by a homeomorphism \(\widehat{f}\) on a (compact) inverse limit space \(\widehat{X}=\{(\dots,x_{-1},x_0):f(x_{i})=x_{i+1}\}\) with the usual metric. For unimodal maps \(\widehat{X}\) may be embedded in \(\mathbb{R}^2\) [\textit{R. H. Bing}, Duke Math. J. 18, 653-663 (1951; Zbl 0043.16804); \textit{M. Misiurewicz}, Fundam. Math. 125, 23-40 (1985; 587.58032)]. The author proves that: If \(f\) is \(C^r\), \(r\geq 0\) then there exists a compact \(A\subset W\subset \mathbb{R}^2\), \(W\) a topological disk, and a map \(g:W\to W\) such that \(g(A)=A\), \((A,g)\) is topologically conjugate to \((\widehat{X},\widehat{f})\), \(g\) is Lipschitz on \(A\) and \(C^r\) on \(W\smallsetminus A\), \(A\) attracts some open neighbourhood of itself. Moreover, the embedding is sufficiently nice to preserve some measure theoretical properties of \(f\), in particular some pathological behaviour of physical (SBR)-measures. The proofs are mainly based on symbolic dynamics of \(f\) (kneading sequences).
scientific article; zbMATH DE number 1340383