Several questions of metric number theory can be successfully handled by the application of ergodic theory. Such an application is possible if an appropriate dynamical system (called fibred system) can be constructed which displays some nice ergodic properties. In this book the basic notion of a fibred system is considered. Most of these systems are connected with f-expansions i.e. generalizations of decimal expansions and continued fractions. There is ample treatment of multidimensional continued fractions including the recent applications of subadditive ergodic theorems to Diophantine approximation. Next, basic properties like ergodicity, conservativity, and existence of invarient measures are discussed. Here several auxiliary constructions as first return map or jim transformation are useful. As a further basic tool the Ruelle-Frobenius-Perron transfer operator is considered.
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Für höhere Schule und Studium
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ISBN-13
978-0-19-853488-4 (9780198534884)
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Schweitzer Klassifikation
1: Basic definitions. 2: Series expansions. 3: Continued fractions. 4: Algorithms of 'cotangent type'. 5: Unimodal maps and piecewise property E-maps. 6: Multidimensional continued fractions. 7: Maps on the real axis. 8: f-expansions. 9: Ergodicity. 10: Examples of non-ergodic systems. 11: Distribution theorems for Oppenheim series. 12: Invariant measures. 13: Kuzmin's equation. 14: The Parry-Daniels maps. 15: Renyi's condition. 16: Auxiliary measures. 17: First return map. 18: Jump transformations I. 19: Jump transformations II. 20: Invariant measures for maps with indifferent fixed points. 21: Dual algorithms. 22: Piecewise fractional linear maps. 23: Ergodocity and invariant measures for multidimensional continued fractions. 24: Natural extensions and Diophantine problems. 25: The Kuzmin-Frobenius-Perron operator. 26: The case of continued fractions. 27: The Lasota-Yorke approach. 28: Distortion and other functionals. 29: Some more results on the Kuzmin operator. 30: The quadratic map. 31: Exponents of convergence for multidimensional continued fractions. References
