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DIMENSION ESTIMATES IN SMOOTH DYNAMICS: A SURVEY OF RECENT RESULTS
"... Abstract. We survey recent results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesti ..."
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Abstract. We survey recent results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and nontrivial developments, only the case of conformal dynamics is completely understood. On the other hand, the study of the dimension of invariant sets of nonconformal maps unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have only a complete understanding of a few classes of invariant sets of nonconformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that numbertheoretical properties do not influence the dimension. Contents
NONCONFORMAL REPELLERS AND THE CONTINUITY OF PRESSURE FOR MATRIX COCYCLES
"... ABSTRACT. The pressure function P (A, s) plays a fundamental role in the calculation of the dimension of “typical ” selfaffine sets, whereA = (A1,..., Ak) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A ..."
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ABSTRACT. The pressure function P (A, s) plays a fundamental role in the calculation of the dimension of “typical ” selfaffine sets, whereA = (A1,..., Ak) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical ” selfaffine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general subadditive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles. 1.
DIMENSION ESTIMATES IN SMOOTH DYNAMICS: A SURVEY
"... Abstract. We survey a collection of results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many i ..."
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Abstract. We survey a collection of results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and nontrivial developments, only the case of conformal dynamics is completely understood. On the other hand, the study of the dimension of invariant sets of nonconformal maps unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have only a complete understanding of a few classes of invariant sets of nonconformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that numbertheoretical properties do not influence the dimension. Contents