Fractal structure of hyperbolic Lipschitzian dynamical systems
In this article, dynamical systems with a Lipschitzian structure are investigated. In the first part of the paper, some properties of hyperbolic Lipschitzian mappings are proved in the terms of the Lipschitzian constants. The main result is the theorem on the semiconjugation of an invertible hyperbolic Lipschitzian dynamical system admiting a pre-Markov partition and the corresponding Markov shift. This result is established in the case there exists some positive constants satisfying the nonlinear system of inequalities. In the second part of the paper, the author suggests a new approach to estimating the Kolmogorov and the Hausdorff dimension of the set of nonwandering points. The proof of these estimations relies on the generalization of the classical theorem on the dimension of the attractor of the matrix iteration system. The generalization is that the contracting condition is replaced by a weaker assumption of contraction along the simple circle.
scientific article; zbMATH DE number 6097099