Hausdorff and upper box dimension estimate of hyperbolic recurrent sets
Let \(N\geq 2\) and \(\{S_{ij} : 1\leq i,j\leq N\}\) be a system of hyperbolic maps defined on a nonempty compact metric space \(X\) in the sense that \(s_{ij}d(x, y)\leq d(S_{ij}(x), S_{ij}(y))\leq \bar{s}_{ij}d(x, y)\) for all \(x,y\in X\) where \(0<s_{ij}\leq \bar{s}_{ij}< 1\) and \(1\leq i, j\leq N\). Let \(M =(m_{ij} )_{1\leq i,j\leq N}\) be an irreducible Markov transition matrix. The collection \(\{X; S_{ij},m_{ij} : 1\leq i, j\leq N\}\) is called a hyperbolic recurrent iterated function system. Let \(E\) be the attractor of the system. If the system of hyperbolic maps satisfies the open set condition: there exists a nonempty open subset \(U\) of \(X\) such that \(S_{ij}(U)\subset U\) for all \(1\leq i, j\leq N\), and \(S_{ij}(U)\bigcap S_{ik}(U) = \emptyset\) for all \(1\leq i, j, k\leq N \) with \(j\neq k\). Let \(h\) and \(H\) be the root of the following equations \[ \|(m_{ij}s_{ij}^{h})_{1\leq i,j\leq N}\| = 1\,\, \text{and}\,\, \|(m_{ij}s_{ij}^{H})_{1\leq i,j\leq N}\| = 1 \] respectively. This paper proves that \(h\leq \dim E\leq H\) and \(0 < \mathcal{H}^h(E)\leq \mathcal{P}^h(E)\) and \(\mathcal{H}^H(E)\leq \mathcal{P}^H(E)<\infty\), where \(\dim E\) stands for the Hausdorff dimension, packing dimension, lower and upper Box dimension of \(E\) respectively, \( \mathcal{H}^s(E)\) and \(\mathcal{P}^s(E)\) denote the \(s\)-dimensional Hausdorff measure and \(s\)-dimensional packing measure of \(E\) respectively.
hyperbolic recurrent iterated function systems
scientific article; zbMATH DE number 6361307