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Includes select papers presented at the Advanced Instructional School on Ergodic Theory and Dynamical Systems
Discusses important and results on ergodic theory and dynamical systems
Focuses on applications in areas in number theory and representation theory
Real Dynamics
Introduction and Preliminaries from Topological Dynamics
Preliminaries
Attracting Fixed Point
Banach's Contraction Mapping Theorem
Various Versions of Attraction
Examples
Topological Transitivity
Five Views of Topological Transitivity
Five Proofs of Topological Transitivity of the Tent Map
Three Ingredients of Chaos
T, DP and SDIC
T, DP &amp NOT SDIC
T, NOT DP &amp SDIC
T, NOT DP &amp NOT SDIC
NOT T, DP &amp SDIC
NOT T, DP &amp NOT SDIC
NOT T, NOT DP &amp SDIC
NOT T, NOT DP &amp NOT SDIC
Chaos For Interval Maps
For Interval Maps Transitivity Implies Chaos
T &amp DP -3mu SDIC
Some Consequences of Intermediate Value Theorem in Dynamics
Immediate Applications
Sarkovskii's Theorem: A Statement
Digraphs of Cycles
Use of digraphs in the proof of Sarkovskii's theorem
Doubling periods
Use of doubling periods in the converse of Sarkovskii's Theorem
Proofs of Some Theorems Used
Notes &amp Exercises
Topological Dynamics
G-Spaces
Minimal Systems
Multiple Recurrence and Van Der Waerden's Theorem
Enveloping Semigroups
Proximal and Distal
Topological Transitivity and Mixing
Basic Ergodic Theory
Measure Theoretic Preliminaries
Measurable Functions and Transformations
Hausdorff Measures
Recurrence and Ergodic Theorems
Recurrence
Birkhoff Ergodic Theorem and the Notion of Ergodicity
Geodesic Flows on Closed Surfaces
Isometries and Geodesics of mathbbH2
Hopf's Proof of Ergodicity
Symbolic Dynamics
Basic Concepts
Entropy
Computations of Entropy
Entropy of Shifts
Entropy of Translations
Tilings
3-Dot Shifts
Complex Dynamics
Some Preliminaries from Complex Analysis and Motivation
Normal Families and Dichotomy of mathbbP1
Rational Maps with Empty Fatou Set
Some Properties of the Julia Set
Local Analysis Near a Fixed Point
Brolin's Theorem
What Happens in Higher Dimensions?
Topics in Homogeneous Dynamics and Number Theory
Homogeneous Dynamics
Diophantine Approximation
On the Distribution of Approximates
The EST Distribution
Spiraling of Approximates
Diophantine Approximation in Number Fields
A Projective Duffin Schaeffer Theorem
The Hyperbolic Picture
On Certain Unusual Large Subsets Arising as Winning Sets of Some Games
Schmidt's ( α,β)-Game
Largeness of Winning Sets
Large Sets Involved in Diophantine Approximation
Large Sets in Geometry and Dynamics
Winning Sets in mathbbRd
Toral Automorphisms
Hyperbolic Geometry
Further Generalisations and Applications
Strong and Absolute Winning Sets
Winning Sets on Lie Groups
Badly Approximable Numbers in Closed Subsets