Includes select papers presented at the Advanced Instructional School on Ergodic Theory and Dynamical Systems
Discusses important concepts and results on ergodic theory and dynamical systems
Focuses on applications in areas in number theory and representation theory
Author(s): Anima Nagar, Riddhi Shah, Shrihari Sridharan
Series: Texts and Readings in Mathematics, 79
Publisher: Springer
Year: 2022
Language: English
Pages: 189
City: Singapore
Preface
Contents
Real Dynamics
1 Introduction and Preliminaries from Topological Dynamics
1.1 Introduction
1.2 Preliminaries
2 Attracting Fixed Point
2.1 Banach's Contraction Mapping Theorem
2.2 Various Versions of Attraction
2.3 Examples
3 Topological Transitivity
3.1 Five Views of Topological Transitivity
3.2 Five Proofs of Topological Transitivity of the Tent Map
4 Three Ingredients of Chaos
4.1 T, DP and SDIC
4.2 T, DP & NOT SDIC
4.3 T, NOT DP & SDIC
4.4 T, NOT DP & NOT SDIC
4.5 NOT T, DP & SDIC
4.6 NOT T, DP & NOT SDIC
4.7 NOT T, NOT DP & SDIC
4.8 NOT T, NOT DP & NOT SDIC
5 Chaos For Interval Maps
5.1 For Interval Maps Transitivity Implies Chaos
5.2 T & DP -3mu SDIC
6 Some Consequences of Intermediate Value Theorem in Dynamics
6.1 Immediate Applications
6.2 Sarkovskii's Theorem: A Statement
6.3 Digraphs of Cycles
6.4 Use of digraphs in the proof of Sarkovskii's theorem
6.5 Doubling periods
6.6 Use of doubling periods in the converse of Sarkovskii's Theorem
7 Proofs of Some Theorems Used
8 Notes & Exercises
References
Topological Dynamics
1 G-Spaces
2 Minimal Systems
3 Multiple Recurrence and Van Der Waerden's Theorem
4 Enveloping Semigroups
5 Proximal and Distal
6 Topological Transitivity and Mixing
7 Summary
References
Basic Ergodic Theory
1 Introduction
2 Measure Theoretic Preliminaries
2.1 Measurable Functions and Transformations
2.2 Hausdorff Measures
3 Recurrence and Ergodic Theorems
3.1 Recurrence
3.2 Birkhoff Ergodic Theorem and the Notion of Ergodicity
4 Geodesic Flows on Closed Surfaces
4.1 Isometries and Geodesics of mathbbH2
4.2 Hopf's Proof of Ergodicity
References
Symbolic Dynamics
1 Introduction
2 Basic Concepts
3 Entropy
4 Computations of Entropy
4.1 Entropy of Shifts
4.2 Entropy of Translations
5 Tilings
6 3-Dot Shifts
References
Complex Dynamics
1 Introduction
2 Some Preliminaries from Complex Analysis and Motivation
3 Normal Families and Dichotomy of mathbbP1
4 Rational Maps with Empty Fatou Set
5 Some Properties of the Julia Set
6 Local Analysis Near a Fixed Point
7 Brolin's Theorem
8 What Happens in Higher Dimensions?
References
Topics in Homogeneous Dynamics and Number Theory
1 Introduction
1.1 Homogeneous Dynamics
1.2 Diophantine Approximation
2 On the Distribution of Approximates
2.1 The EST Distribution
2.2 Spiraling of Approximates
3 Diophantine Approximation in Number Fields
4 A Projective Duffin Schaeffer Theorem
5 The Hyperbolic Picture
References
On Certain Unusual Large Subsets Arising as Winning Sets of Some Games
1 Introduction
2 Schmidt's ( α,β)-Game
3 Largeness of Winning Sets
4 Large Sets Involved in Diophantine Approximation
5 Large Sets in Geometry and Dynamics
5.1 Winning Sets in mathbbRd
5.2 Toral Automorphisms
5.3 Hyperbolic Geometry
6 Further Generalisations and Applications
6.1 Strong and Absolute Winning Sets
6.2 Winning Sets on Lie Groups
6.3 Badly Approximable Numbers in Closed Subsets
References
