Topics in Dynamics and Ergodic Theory

This collection of survey papers by leading researchers in ergodic theory and low-dimensional and topological dynamics comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics. These include: the role and usefulness of ultrafilters in ergodic theory, topological dynamics and Ramsey theory; topological aspects of kneading theory together with an analogous 2-dimensional theory called pruning; the dynamics of Markov odometers, Bratteli-Vershik diagrams and orbit equivalence of non-singular automorphisms; geometric proofs of Mather's connecting and accelerating theorems; recent results in one dimensional smooth dynamics; periodic points of nonexpansive maps; arithmetic dynamics; the defect of factor maps; entropy theory for actions of countable amenable groups.

1. Introductory talk at the opening of the conference Anatole M. Stepin; 2. Minimal idempotents and ergodic Ramsey theory Vitaly Bergelson; 3. Symbolic dynamics and topological models in dimensions 1 and 2 André de Carvalho and Toby Hall; 4. Markov odometers Anthony H. Dooley; 5. Geometric proofs of Mather's connecting and accelerating theorems Vadim Kaloshin; 6. Structural stability in one dimensional dynamics Oleg Kozlovski; 7. Periodic points of nonexpansive maps: a survey Bas Lemmens; 8. Arithmetic dynamics Nikita Sidorov; 9. The defect of factor maps and finite equivalence of dynamical systems Klaus Thomsen; 10. Actions of amenable groups Benjamin Weiss.

This book contains a collection of survey papers by leading researchers in ergodic theory, low-dimensional and topological dynamics and it comprises nine chapters on a range of important topics.