Dimension Theory in Dynamical Systems: Contemporary Views and Applications
The principles of symmetry and self-similarity structure nature's most beautiful creations. For example, they are expressed in fractals, famous for their beautiful but complicated geometric structure, which is the subject of study in dimension theory. And in dynamics the presence of invariant fractals often results in unstable "turbulent-like" motions and is associated with "chaotic" behavior.
In this book, Yakov Pesin introduces a new area of research that has recently appeared in the interface between dimension theory and the theory of dynamical systems. Focusing on invariant fractals and their influence on stochastic properties of systems, Pesin provides a comprehensive and systematic treatment of modern dimension theory in dynamical systems, summarizes the current state of research, and describes the most important accomplishments of this field.
Pesin's synthesis of these subjects of broad current research interest will be appreciated both by advanced mathematicians and by a wide range of scientists who depend upon mathematical modeling of dynamical processes.
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Appendix assume balls of radius Bar2 Barreira Borel finite measure Bowen's equation C-structure coding map coincide compact conformal repellers consider constant cylinder set define denote diametrically regular diffeomorphism dimBF dimBZ dime dimension and box dimension and lower dimension theory disjoint dynamical systems equilibrium measure corresponding ergodic measure example expanding map finite or countable finite type full shift Gibbs measure Given Hausdorff dimension hence Holder continuous function homeomorphism hyperbolic set i-almost implies inequality infimum infimum is taken integer intersection invariant measure invariant set Lemma limit set F lower and upper Lyapunov exponents map f Markov partition measure-theoretic entropy metric space Moran cover multifractal analysis obtain pointwise dimension Proposition ratio coefficients respectively root of Bowen's satisfy Condition Section set function set Z C X subset subshift of finite symbolic dynamical symbolic dynamical system topological entropy topological pressure topologically mixing unique root upper box dimensions upper Caratheodory capacities