Dynamics and Numbers.

Cover -- Title page -- Contents -- Preface -- Bratteli diagrams: Structure, measures, dynamics -- 1. Introduction -- 2. Fundamentals of Cantor dynamics and Bratteli diagrams -- 2.1. Cantor dynamics -- 2.2. Bratteli diagrams -- 2.3. Ordered Bratteli diagrams and Vershik maps -- 3. Bratteli-Vershik representations of Cantor dynamical systems and orbit equivalence -- 3.1. Bratteli-Vershik model of a Cantor dynamical system -- 3.2. Orbit equivalence and full groups -- 4. Bratteli diagrams that admit a continuous Vershik map -- 5. Stationary Bratteli diagrams -- 5.1. Substitution dynamical systems -- 5.2. Invariant measures on stationary diagrams and their supports -- 5.3. Good measures on stationary Bratteli diagrams -- 5.4. Complexity and orbit equivalence -- 6. Finite rank Bratteli diagrams -- 7. Measures and subdiagrams -- References -- The combinatorial Mandelbrot set as the quotient of the space of geolaminations -- Introduction -- 1. Preliminaries -- 2. Limit geolaminations and their properties -- 3. The Mandelbrot set as the quotient of the space of quadratic limit geolaminations -- References -- Shearer's inequality and infimum rule for Shannon entropy and topological entropy -- 1. Introduction -- 2. Subadditivity and related notions -- 3. The infimum rule -- 4. Shannon entropy -- 5. Beyond amenability -- 6. Topological entropy -- References -- The Nielsen and Reidemeister theories of iterations on infra-solvmanifolds of type ( ) and poly-Bieberbach groups -- 1. Introduction -- 2. Preliminaries -- 3. Poly-Bieberbach groups -- 4. The Nielsen theory of iterations on an infra-solvmanifold of type \R -- 5. Reidemeister numbers ( ^{ }) -- 6. Radius of convergence of _{ }( ) -- 7. Asymptotic behavior of the sequence { ( ^{ })} -- 8. Periodic [ ]-orbits -- 9. Heights of -- References -- Some remarks on modified power entropy -- 1. Introduction.

2. Some thoughts on slow entropies -- 3. Power entropy, modified power entropy and amorphic complexity -- 4. Modified power entropy and topological entropy -- 5. A counterexample to the existence of a variational principle for modified power entropy -- 6. Modified power entropy of regular almost 1-1 extensions -- 7. An example of an irregular Toeplitz flow of low complexity -- 8. Conclusions and open questions -- References -- Recurrent sets, entropy and independence -- 1. Introduction -- 2. Preliminaries -- 3. Proof of Main Results -- 4. Further discussions -- References -- Loops of transitive interval maps -- 1. Introduction -- 2. Auxiliary results on transitivity -- 3. Spaces \tcs₁ and \tcs₂ -- 4. Basic loops -- 5. Contracting the basic loop of order 2 -- 6. Contracting basic loops of higher order -- 7. Contracting the basic loop of order 1 -- Acknowledgments -- References -- A panorama of specification-like properties and their consequences -- Acknowledgements -- 1. Basic definitions and notation -- 2. Specification property -- 3. Weak specification -- 4. Almost specification -- 5. Almost and weak specification -- 6. Approximate product property -- 7. Specification and shadowing -- 8. Symbolic dynamics -- References -- Counting dynamical systems over finite fields -- 1. Introduction -- 2. Previous results -- 3. Main results -- 4. Comments and questions -- Acknowledgements -- References -- Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations -- 1. Introduction -- 2. Preliminaries -- 3. Weighted discretizations and transfer operators -- 4. The slow systems and automorphic forms -- 5. The fast systems and the Selberg zeta functions -- 6. Symmetry reduction -- 7. Billiard flows -- References -- The aperiodic complexities and connections to dimensions and Diophantine approximation.

1. Introduction and main results -- 2. Quantitatively Aperiodic and Recurrent Orbits -- 3. Examples -- References -- Dynamical systems of non-algebraic origin: Fixed points and orbit lengths -- 1. Introduction -- 2. Dynamics of Fermat Quotients -- 3. Dynamics of the Exponential Function -- 4. Dynamics of the Self-Exponential Function -- 5. Other Dynamical Systems -- 6. Comments -- Acknowledgements -- References -- Halving dynamical systems -- 1. Introduction -- 2. Closed orbits -- 3. Closed orbits and topological factors -- 4. Shortening, Surviving, and Gluing Orbits -- 5. The basic lemma -- 6. Growth in closed orbits -- 7. Dynamical zeta functions -- 8. Concluding remarks and questions -- References -- Chaotic behavior of group actions -- Acknowledgements -- 1. Introduction -- 2. Weak mixing in countable discrete group actions -- 3. Preliminaries for entropy of amenable group actions -- 4. Homoclinic equivalence relation in a shift of finite type for amenable group actions with positive entropy -- 5. Weak mixing in amenable group actions with positive entropy -- References -- Back Cover.

This volume contains a collection of survey and research articles from the special program and international conference on Dynamics and Numbers held at the Max-Planck Institute for Mathematics in Bonn, Germany in 2014. The papers reflect the great diversity and depth of the interaction between number theory and dynamical systems and geometry in particular. Topics covered in this volume include symbolic dynamics, Bratelli diagrams, geometry of laminations, entropy, Nielsen theory, recurrence, topology of the moduli space of interval maps, and specification properties.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.