This book offers a comprehensive overview of dimension theory of hyperbolic flows. It includes a detailed discussion of major open problems in the area.
First comprehensive exposition of dimension theory of hyperbolic flows Includes an overview of dimension theory and multifractal analysis Includes a detailed discussion of major open problems in the area
1.Introduction.- Part I Basic Notions .- 2.Suspension Flows.- 3.Hyperbolic Flows.- 4.Pressure and Dimension.- Part II Dimension Theory .- 5.Dimension of Hyperbolic Sets.- 6.Pointwise Dimension and Applications.- Part III Multifractal Analysis .- 7.Suspensions over Symbolic Dynamics.- 8.Multifractal Analysis of Hyperbolic Flows.- Part IV Variational Principles .- 9.Entropy Spectra.- 10.Multidimensional Spectra.- 11.Dimension Spectra.- References.- Index.
The dimension theory of dynamical systems has progressively developed, especially over the last two decades, into an independent and extremely active field of research. Its main aim is to study the complexity of sets and measures that are invariant under the dynamics. In particular, it is essential to characterizing chaotic strange attractors. To date, some parts of the theory have either only been outlined, because they can be reduced to the case of maps, or are too technical for a wider audience. In this respect, the present monograph is intended to provide a comprehensive guide. Moreover, the text is self-contained and with the exception of some basic results in Chapters 3 and 4, all the results in the book include detailed proofs.
The book is intended for researchers and graduate students specializing in dynamical systems who wish to have a sufficiently comprehensive view of the theory together with a working knowledge of its main techniques. The discussion of some open problems is also included in the hope that it may lead to further developments. Ideally, readers should have some familiarity with the basic notions and results of ergodic theory and hyperbolic dynamics at the level of an introductory course in the area, though the initial chapters also review all the necessary material.