Ergodic theory of fibred systems and metric number theory
The application of ergodic theory to numerous problems in metric number theory--possible when a fibred system is constructed--has yielded promising results. This book details the basic notion of fibred systems, most of which are connected with f-expansions. Topics include multidimensional continued fractions (such as the recent applications of subadditive ergodic theorems to Diophantine approximation), ergodicity, conservativity, the existence of invariant measures, and the Ruelle-Freobenius-Perron transfer operator. Containing a wealth of information previously unavailable in book form, Ergodic Theory of Fibred Systems and Metric Number Theory will be welcomed by advanced students and researchers in chaos theory, number theory, and physics.
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absolutely continuous admits a finite admits an invariant An+i B(ki Benford's law Bn+i Borel sets bounded variation Brun's algorithm Chapter clearly conservative consider constant continued fractions convergence Corollary define Definition Let denote Diophantine Diophantine approximation dual algorithm dynamical systems equation ergodic and admits ergodic theory examples fc=i fibred system finite invariant measure following conditions full cylinders function Furthermore given Hence Hofbauer implies indifferent fixed point individual ergodic theorem infinitely integral interval invariant density inverse ip(x iteration Jacobi-Perron algorithm Jacobian Jager jump transformation Keller Lebesgue measure Lyapunov exponents map Tx Markov Markov property Math measurable function metrical theory Nakada Number Theory obtain oo oo partition Proof properties real number relation Remark Renyi's condition result return map Schweiger sequence Sylvester's series TB(k Thaler Theorem Let wandering set Xi Xi Xi Xi Xi Yuri