Iterated function systems and the code space
Let \(\{1,2,\ldots, N\}\) be a finite alphabet and \(\Sigma=\prod_{i=1}^\infty\{1,2,\ldots, N\}\) the associated code space. This space is the domain of the address map \(g\) associated to an iterated function system (IFS) of similitudes \(\{f_{1}, f_{2},\ldots,f_{N}\}\), with \(f_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) of similarity factor \(0<c_{i}<1\), \(i=1,\dots,N\). Namely, if \(K\) is the selfsimilar set associated to the IFS, then \(g:\Sigma\to K\) is defined by: \[ g(i_{1}, i_{2},\ldots, i_{n},\ldots)=\lim_{n\to\infty}f_{i_{1}}\circ\cdots\circ f_{i_{n}}. \] In this paper the authors define an ultrametric on the code space and establish a natural connection between the corresponding metric space and the invariant set \(K\) of the IFS having as similarity factors the constants used in the definition of the ultrametric. Independently on an IFS one considers a set of numbers \(0<c_{i}<1\), \(i=1,\dots,N\). Then the ultrametric \(\rho\) defined on the code space \(\Sigma\) is given by \(\rho(i_{j},k_{j})=c_{i_{1}}\cdots c_{i_{n}}\), if the two sequences agree up to index \(n\) and disagree at index \(n+1\); \(\rho(i_{j},k_{j})=0\) if the two sequences are identical, and \(\rho(i_{j},k_{j})=1\) if they differ in the first element of the sequence. One of the results of the paper states that if the constants \(c_{i}\) satisfy \(0<c_{i}<{1\over{2}}\), \(i\in\{1,2\ldots,N\}\), then there is a bi-Lipschitz embedding \(g\) of the metric space \((\Sigma,\rho)\) in some \(\mathbb{R}^{n}\), where \(n\) is any integer satisfying \(2^{n}\geq N\). Moreover the embedding \(g\) is such that \(g(\Sigma)\) is a selfsimilar set defined by an IFS of \(N\) similitudes, of similarity factors \(\{c_{1},\ldots, c_{N}\}\), and satisfying the open set condition. A second result consists in a formula for the Hausdorff dimension of a subshift of finite type, \(\Sigma_{B}\), of the metric space \((\Sigma,\rho)\) (\(B=(b_{ij})\) is the transition matrix defining the subshift). Namely, it is shown that the Hausdorff dimension of \(\Sigma_{B}\) is equal to \(\beta\), where \(\beta\) is the unique value of \(s\) such that the matrix \(M(s)\) of entries \(m_{ij}(s)=b_{ij}c_{j}^{s}\), \(i,j=1,\dots, N\), has largest real eigenvalue \(\lambda=1\). Finally the authors have shown that the image of the bi-Lipschitz embedding of the code space in some \(\mathbb{R}^{n}\) can be the maximal compact invariant set of a smooth map \(F:\mathbb{R}^{n}\to\mathbb{R}^{n}\). The map \(F\) restricted to \(g(\Sigma)\) is equivalent to a power of the shift map, and the Hausdorff dimension of the invariant set, as well as the Lyapunov exponent of \(F\) at various points of this set, and the topological entropy of \(F\) can be also computed.
scientific article; zbMATH DE number 1787550
scientific article