Dynamics : topology and numbers : Conference on dynamics : topology and numbers, July 2-6, 2018, Max Planck Institute for Mathematics, Bonn, Germany

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. Cover 1 Title page 4 Contents 8 Preface 10 The life and mathematics of Sergiĭ Kolyada 12 1. On the life of Sergiĭ Kolyada 12 2. On the mathematical work of Sergiĭ Kolyada 14 Papers by Sergiĭ Kolyada 22 Books by Sergiĭ Kolyada 24 Publications co-edited by Sergiĭ Kolyada 24 Other references 25 Recollections about Sergiĭ Kolyada 26 Sergiy and the MPIM 32 References 35 Homotopy types and geometries below Spec(Z) 38 1. Brief summary and plan of exposition 38 2. Roots of unity as Weil numbers 42 3. The Bost–Connes system and the Grothendieck ring 44 4. From Rings to Spectra 49 5. Expectation values, motivic measures, and zeta functions 60 6. Dynamical \F1-structures and the Bost–Connes algebra 62 References 65 Dynamical zeta functions of Reidemeister type and representations spaces 68 1. Introduction 68 2. Dynamical zeta functions and representations spaces 70 3. Connection with Reidemeister Torsion 76 4. Reduction to subgroups and quotient groups 78 5. Pólya –Carlson dichotomy for Reidemeister zeta function 83 References 90 Rigorous dimension estimates for Cantor sets arising in Zaremba theory 94 1. Introduction 94 2. Preliminaries 97 3. Bounding dimension determinant coefficients 99 4. The Hausdorff dimension of E_{{2,4,6,8,10}} is greater than 1/2 101 5. The Hausdorff dimension of E_{{1,2,3,4,5}} 107 6. The Hausdorff dimension of E_{{1,2,3,4,5,6}} 113 References 117 Volume growth for infinite graphs and translation surfaces 120 1. Introduction 120 2. Infinite Graphs 121 3. Countable Matrices 123 4. Complex functions 124 5. Proof of Theorem 2.1 126 6. Translation surfaces 127 References 133 Dynamically affine maps in positive characteristic 136 1. Introduction 136 2. Generalities 143 3. Introduction of the general hypotheses 145 4. Proofs of Theorems 3.5 and 3.6 147 5. Discussion of the hypotheses 153 Appendix A. Radius of convergence of ζ_{f} for dynamically affine maps f 159 Appendix B. Explicit computation of tame zeta functions for some dynamically affine maps on \PP1 162 Acknowledgments 164 References 164 Special α-limit sets 168 1. Introduction 168 2. General case 170 3. Interval maps 173 4. Examples 178 References 183 Equicontinuity of minimal sets for amenable group actions on dendrites 186 1. Introduction 186 2. Preliminaries 187 3. Proof of the main theorem 189 References 190 On weak rigidity and weakly mixing enveloping semigroups 192 1. Introduction 192 2. Recurrence 193 3. Some obstructions to WM of E(X,T) 197 4. The horocycle flow 199 References 200 The inhomogeneous Sprindžhuk conjecture over a local field of positive characteristic 202 1. Introduction 202 Acknowledgments 204 2. Homogeneous and Inhomogeneous Diophantine exponents 205 3. Good and nonplanar maps 206 4. Transference principles and lower bounds 207 5. The Transference principle of Beresnevich-Velani 208 6. Proof of Theorem 1.1 209 7. Further directions 210 References 212 Dynamical generation of parameter laminations 216 Introduction 216 1. Majors and minors 223 2. Derived minors, children, and offsprings: proof of Theorem A 230 3. Coexistence and tuning 233 4. Almost non-renormalizable minors: proof of Theorem B 235 References 238 Multi-sensitivity, multi-transitivity and Δ-transitivity 242 1. Introduction 242 2. Preliminaries 244 3. Sensitivity and transitivity with respect to a vector 245 4. Weak mixing and Δ-transitivity 250 5. Multi-ergodicity 252 Acknowledgments 254 References 254 Convergence of zeta functions for amenable group extensions of shifts 256 1. Introduction 256 2. Graphs 260 3. Weighted Zeta Functions and Metric Graphs 262 4. Traces 264 5. Proof of Theorem 1.2 266 References 268 Invariant measures for Cantor dynamical systems 270 1. Introduction 271 2. Basics on Cantor dynamics and Bratteli diagrams 273 2.1. Cantor dynamical systems 273 2.2. Languages on finite alphabets and complexity 274 2.3. Ordered Bratteli diagrams and Vershik maps 276 3. Invariant measures on Bratteli diagrams 280 3.1. Simplices, stochastic incidence matrices, examples 280 3.2. Subdiagrams and measure extension (finite and infinite measures) 285 4. Uniquely ergodic Cantor dynamical systems 286 4.1. Minimal uniquely ergodic homeomorphisms in symbolic dynamics 287 4.2. Finite rank Bratteli diagrams and general case 288 4.3. Examples 290 5. Finitely ergodic Cantor dynamical systems 292 5.1. Finitely ergodic subshifts 292 5.2. Stationary Bratteli diagrams 294 5.3. Finite rank Bratteli diagrams 295 5.4. Examples 299 6. Infinite rank Cantor dynamical systems 300 6.1. A class of Bratteli diagrams of infinite rank 300 6.2. Examples 301 References 303 Periods of abelian differentials and dynamics 308 1. Introduction 308 2. Geometric preliminaries 312 3. Ratner’s Theorem 317 4. The semisimple case 319 5. The non-semisimple case 322 6. Meromorphic differentials 323 References 325 Crossed renormalization of quadratic polynomials 328 1. Introduction 329 2. Background on quadratic polynomials 330 2.1. The Mandelbrot set and Julia sets 330 2.2. Dynamic rays, parameter rays, and equipotentials 330 2.3. Polynomial-like maps 333 2.4. Renormalization 335 3. Crossed renormalization: the immediate case 337 3.1. The principal construction 338 3.2. The boundary of the renormalization locus 340 3.3. A homeomorphism from C_{p,q}n to the p/q-limb 344 3.4. Our construction is complete 347 3.5. Crossed tuning 348 3.6. Internal addresses 349 3.7. Puzzles and tableaux 353 4. Crossed renormalization: the general case 354 References 357 Back Cover 360 Contains the proceedings of the conference Dynamics: Topology and Numbers, held in July 2018. 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.