Dynamics : topology and numbers : Conference dynamics : topology and numbers, July 2-6, 2018, Max Planck Institute for Mathematics, Bonn, Germany / Pieter Moree [and three others], editors.

Format:

Book

Author/Creator:

Moree, Pieter, 1965- editor.

Contributor:

Moree, Pieter, 1965- editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 744

Contemporary mathematics, 744 0271-4132

Language:

English

Subjects (All):

Dynamics--Congresses.

Dynamics.

Physical Description:

1 online resource (x, 347 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

[Place of publication not identified] : American Mathematical Society, [2020]

Language Note:

English

Summary:

This volume contains the proceedings of the conference Dynamics: Topology and Numbers, held from July 2-6, 2018, at the Max Planck Institute for Mathematics, Bonn, Germany. The papers cover diverse fields of mathematics with a unifying theme of relation to dynamical systems. These include arithmetic geometry, flat geometry, complex dynamics, graph theory, relations to number theory, and topological dynamics. The volume is dedicated to the memory of Sergiy Kolyada and also contains some personal accounts of his life and mathematics.

Contents:

Cover

Title page

Contents

Preface

The life and mathematics of Sergiĭ Kolyada

1. On the life of Sergiĭ Kolyada

2. On the mathematical work of Sergiĭ Kolyada

Papers by Sergiĭ Kolyada

Books by Sergiĭ Kolyada

Publications co-edited by Sergiĭ Kolyada

Other references

Recollections about Sergiĭ Kolyada

Sergiy and the MPIM

References

Homotopy types and geometries below (ℤ)

1. Brief summary and plan of exposition

2. Roots of unity as Weil numbers

3. The Bost-Connes system and the Grothendieck ring

4. From Rings to Spectra

5. Expectation values, motivic measures, and zeta functions

6. Dynamical \F₁-structures and the Bost-Connes algebra

Dynamical zeta functions of Reidemeister type and representations spaces

1. Introduction

2. Dynamical zeta functions and representations spaces

3. Connection with Reidemeister Torsion

4. Reduction to subgroups and quotient groups

5. Pólya -Carlson dichotomy for Reidemeister zeta function

Rigorous dimension estimates for Cantor sets arising in Zaremba theory

2. Preliminaries

3. Bounding dimension determinant coefficients

4. The Hausdorff dimension of _{{2,4,6,8,10}} is greater than 1/2

5. The Hausdorff dimension of _{{1,2,3,4,5}}

6. The Hausdorff dimension of _{{1,2,3,4,5,6}}

Volume growth for infinite graphs and translation surfaces

2. Infinite Graphs

3. Countable Matrices

4. Complex functions

5. Proof of Theorem 2.1

6. Translation surfaces

Dynamically affine maps in positive characteristic

2. Generalities

3. Introduction of the general hypotheses

4. Proofs of Theorems 3.5 and 3.6

5. Discussion of the hypotheses.

Appendix A. Radius of convergence of _{ } for dynamically affine maps

Appendix B. Explicit computation of tame zeta functions for some dynamically affine maps on \PP¹

Acknowledgments

Special -limit sets

2. General case

3. Interval maps

4. Examples

Equicontinuity of minimal sets for amenable group actions on dendrites

3. Proof of the main theorem

On weak rigidity and weakly mixing enveloping semigroups

2. Recurrence

3. Some obstructions to WM of ( , )

4. The horocycle flow

The inhomogeneous Sprindžhuk conjecture over a local field of positive characteristic

2. Homogeneous and Inhomogeneous Diophantine exponents

3. Good and nonplanar maps

4. Transference principles and lower bounds

5. The Transference principle of Beresnevich-Velani

6. Proof of Theorem 1.1

7. Further directions

Dynamical generation of parameter laminations

Introduction

1. Majors and minors

2. Derived minors, children, and offsprings: proof of Theorem A

3. Coexistence and tuning

4. Almost non-renormalizable minors: proof of Theorem B

Multi-sensitivity, multi-transitivity and Δ-transitivity

3. Sensitivity and transitivity with respect to a vector

4. Weak mixing and Δ-transitivity

5. Multi-ergodicity

Convergence of zeta functions for amenable group extensions of shifts

2. Graphs

3. Weighted Zeta Functions and Metric Graphs

4. Traces

5. Proof of Theorem 1.2

Invariant measures for Cantor dynamical systems

2. Basics on Cantor dynamics and Bratteli diagrams.

2.1. Cantor dynamical systems

2.2. Languages on finite alphabets and complexity

2.3. Ordered Bratteli diagrams and Vershik maps

3. Invariant measures on Bratteli diagrams

3.1. Simplices, stochastic incidence matrices, examples

3.2. Subdiagrams and measure extension (finite and infinite measures)

4. Uniquely ergodic Cantor dynamical systems

4.1. Minimal uniquely ergodic homeomorphisms in symbolic dynamics

4.2. Finite rank Bratteli diagrams and general case

4.3. Examples

5. Finitely ergodic Cantor dynamical systems

5.1. Finitely ergodic subshifts

5.2. Stationary Bratteli diagrams

5.3. Finite rank Bratteli diagrams

5.4. Examples

6. Infinite rank Cantor dynamical systems

6.1. A class of Bratteli diagrams of infinite rank

6.2. Examples

Periods of abelian differentials and dynamics

2. Geometric preliminaries

3. Ratner's Theorem

4. The semisimple case

5. The non-semisimple case

6. Meromorphic differentials

Crossed renormalization of quadratic polynomials

2. Background on quadratic polynomials

2.1. The Mandelbrot set and Julia sets

2.2. Dynamic rays, parameter rays, and equipotentials

2.3. Polynomial-like maps

2.4. Renormalization

3. Crossed renormalization: the immediate case

3.1. The principal construction

3.2. The boundary of the renormalization locus

3.3. A homeomorphism from _{ , }ⁿ to the / -limb

3.4. Our construction is complete

3.5. Crossed tuning

3.6. Internal addresses

3.7. Puzzles and tableaux

4. Crossed renormalization: the general case

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

This was the fourth conference at MPIM on the theme of dynamical systems and its relationships with the diverse fields of number theory, geometry, topology, ergodic theory, and combinatorics.

ISBN:

9781470454548

1470454548