Dynamics of Circle Mappings

Preface
Contents
List of Symbols
Part I Basic Theory
	1 Rotations
		1.1 Topology and Combinatorics of Rotations
			1.1.1 A Dichotomy
			1.1.2 Sequence of Closest Returns
		1.2 Rotations and Continued Fractions
			1.2.1 Basic Theory of Continued Fractions
			1.2.2 Best Approximations
		1.3 Weyl\'s Equidistribution Theorem
			1.3.1 Equidistribution
			1.3.2 A Simple Application
		1.4 Ergodicity of Irrational Rotations
		Exercises
	2 Homeomorphisms of the Circle
		2.1 Translation and Rotation Numbers
			2.1.1 The Classical Definition
			2.1.2 The Order Definition
			2.1.3 The Measure-Theoretic Definition
			2.1.4 Properties of the Rotation Number
		2.2 Topological Dynamics of Homeomorphisms
			2.2.1 Rational Rotation Number
			2.2.2 Irrational Rotation Number
		2.3 Invariant Measures and Semi-Conjugacies
		Exercises
Part II Diffeomorphisms
	3 Diffeomorphisms: Denjoy Theory
		3.1 The Naive Distortion Lemma
		3.2 Denjoy\'s Theorem
			3.2.1 The C2 Version
			3.2.2 The Bounded Variation Version
		3.3 Denjoy\'s Examples
			3.3.1 The Basic Construction
			3.3.2 Moduli of Continuity
			3.3.3 Further Results
				3.3.3.1 Classification
				3.3.3.2 Which Cantor Sets Are Denjoy?
				3.3.3.3 Hausdorff Dimension
		3.4 Ergodic Properties
			3.4.1 Ergodicity with Respect to Lebesgue Measure
			3.4.2 Zero Lyapunov Exponents
			3.4.3 Further Ergodicity Results
				3.4.3.1 Automorphic Measures
				3.4.3.2 Invariant Distributions
		Exercises
	4 Smooth Conjugacies to Rotations
		4.1 Herman\'s Invariants
		4.2 Small Denominators: Arnold\'s Theorem
			4.2.1 The Linearized Equation
			4.2.2 Non-linear Estimates
			4.2.3 Proof of Arnold\'s Theorem
		4.3 Counterexamples to Linearizability
			4.3.1 One-Parameter Families
			4.3.2 Residual Sets of Non-linearizable Parameters
			4.3.3 Singular Measures and Conjugacies
		4.4 Further Local Theory: The Brjuno Condition
		4.5 Global Theory: Herman–Yoccoz Results and Beyond
		Exercises
Part III Multicritical Circle Maps
	5 Cross-Ratios and Distortion Tools
		5.1 Cross-Ratios
		5.2 The Schwarzian
			5.2.1 Definition
			5.2.2 Koebe\'s Nonlinearity Principle
			5.2.3 The Minimum Principle
		5.3 Distortion and Cross-Ratio Distortion
			5.3.1 Koebe\'s Distortion Principle
			5.3.2 Distortion and the Schwarzian
			5.3.3 Behavior Near Critical Points
		5.4 The Cross-Ratio Inequality
		5.5 A Cancellation Lemma
		Exercises
	6 Topological Classification and the Real Bounds
		6.1 Definition and Examples of Multicritical Circle Maps
			6.1.1 Blaschke Products
			6.1.2 The Arnold Family
		6.2 Topological Classification
			6.2.1 Dynamically Symmetric Intervals
			6.2.2 Proof of Yoccoz\'s Theorem
		6.3 Real a Priori Bounds
			6.3.1 Dynamical Partitions
			6.3.2 The Real Bounds
				6.3.2.1 Comparability of Closest Returns and Beyond
				6.3.2.2 Proof of Theorem 6.3
			6.3.3 On the Notion of Comparability
		6.4 First Consequences
			6.4.1 C1 Bounds
			6.4.2 Sums of Polar Ratios
		6.5 A Negative Schwarzian Property
		6.6 Beau Bounds
		Exercises
	7 Quasisymmetric Rigidity
		7.1 Quasisymmetry and Fine Grids
			7.1.1 A Criterion for Quasisymmetry
			7.1.2 A Criterion for Smoothness
		7.2 Quasisymmetric Conjugacies
		7.3 Almost Parabolic Maps
			7.3.1 Yoccoz\'s Inequality
			7.3.2 Balanced Decompositions
		7.4 Quasisymmetric Rigidity
			7.4.1 More on the Geometry of Dynamical Partitions
				7.4.1.1 Intersecting Atoms Are Comparable
				7.4.1.2 Critical Atoms Are Large
			7.4.2 Building a Suitable Fine Grid
				7.4.2.1 Auxiliary Partitions
				7.4.2.2 Balanced Decompositions of Bridges
				7.4.2.3 The Recursive Scheme
			7.4.3 The Punchline
		Exercises
	8 Ergodic Aspects
		8.1 The Integrability of logDf
		8.2 No Invariant σ-Finite Measures
			8.2.1 The Katznelson Criterion
			8.2.2 Proof of Theorem 8.1
				8.2.2.1 First Step
				8.2.2.2 Second Step
				8.2.2.3 The Punchline
			8.2.3 Negative Schwarzian Redux
				8.2.3.1 Bounded Geometry
				8.2.3.2 Proving that the Schwarzian Is Negative
		8.3 Lyapunov Exponents
			8.3.1 The Collet–Eckmann Condition
			8.3.2 The Key Step
		8.4 Further Ergodic Properties
		8.5 Hausdorff Dimension
		Exercises
	9 Orbit Flexibility
		9.1 Geometric Equivalence of Orbits
			9.1.1 Orbit-Flexibility
			9.1.2 Statement for Unicritical Maps
			9.1.3 Statements for Multicritical Maps
			9.1.4 Centralizers
			9.1.5 Unbounded Geometry
		9.2 Renormalization Trails and Ancestors
		9.3 The Skew Product
			9.3.1 The Fiber Maps
			9.3.2 The Skew Product
		9.4 Proof of Theorem 9.6
		9.5 Even-Type Rotation Numbers
		9.6 Proofs of Theorems 9.1 and 9.2
		9.7 The C∞ Realization Lemma
			9.7.1 Admissible Pairs
		Exercises
Part IV Renormalization Theory
	10 Smooth Rigidity and Renormalization
		10.1 Smooth Rigidity
		10.2 Renormalization of Commuting Pairs
		10.3 A Fundamental Principle
			10.3.1 Main Theorem
			10.3.2 Comparing Orbits of Two Almost Parabolic Maps
			10.3.3 Proof of Theorem 10.4
		10.4 The Cm-Approximation Lemma
		10.5 Counterexamples to C1+α Rigidity
			10.5.1 Saddle-Node Surgery
			10.5.2 The Counterexamples
		Exercises
	11 Quasiconformal Deformations
		11.1 Quasiconformal Homeomorphisms
			11.1.1 The Geometric Definition
			11.1.2 The Analytic Definition
			11.1.3 Measurable Riemann Mapping Theorem
		11.2 A Simple Dynamical Application
		11.3 Holomorphic Approximation Lemma
		Exercises
	12 Lipschitz Estimates for Renormalization
		12.1 Lipschitz Estimates for Controlled Commuting Pairs
		12.2 Standard Families
			12.2.1 Glueing Procedure and Translations
			12.2.2 Standard Families of Commuting Pairs
			12.2.3 Renormalization of Standard Families
		12.3 Orbit Deformations
		12.4 Composition
		12.5 Order
		12.6 Synchronization
		12.7 Lipschitz Estimate
		Exercises
	13 Exponential Convergence: The Smooth Case
		13.1 The Shadowing Property
			13.1.1 Extended Lifts of Critical Circle Maps
			13.1.2 Almost Schwarz Inclusion
			13.1.3 A Bidimensional Glueing Procedure
			13.1.4 Main Perturbation
			13.1.5 The Shadowing Sequence
		13.2 Bounding the Cr-1 Metric
		13.3 Proof of the Exponential Convergence
		13.4 The Attractor of Renormalization
		Exercises
	14 Renormalization: Holomorphic Methods
		14.1 Sullivan\'s Program
		14.2 Holomorphic Commuting Pairs
		14.3 Pull-Back Argument
		14.4 Existence and Limit-Set qc-Rigidity
			14.4.1 Construction of Examples
				14.4.1.1 Limit Set qc-Rigidity
		14.5 Complex Bounds
		14.6 McMullen\'s Dynamic Inflexibility Theorem
		14.7 Proof of Exponential Convergence
		14.8 Hyperbolicity of Renormalization
			14.8.1 Cylindrical and Parabolic Renormalizations
				14.8.1.1 Parabolic Renormalization: Fatou Coordinates
				14.8.1.2 Douady Coordinates
				14.8.1.3 Cylindrical Renormalization
				14.8.1.4 Final Remarks
		Exercises
Epilogue
	Epilogue
A Ergodic Theory of Continued Fractions
	A.1 Expansions as Itineraries
	A.2 The Gauss Measure and Almost Surely Properties
	A.3 Diophantine Approximations Revisited
	Exercises
B Cohomological Equations and Smooth Conjugacies
	B.1 The Cohomological Equation
	B.2 Solving the Cohomological Equation
	B.3 Distortion Estimates
		B.3.1 Distortion Estimates in C1+BV
		B.3.2 Distortion Estimates in C2+α
	B.4 C1 Exponential Convergence
		B.4.1 Further Distortion Estimates
	B.5 The Diophantine Condition
		B.5.1 Proof of Proposition B.3
	B.6 Proof of Theorem 4.11
		B.6.1 Hölder Continuity of the Invariant Density
C A Skew Product Over the Gauss Map
	C.1 An Absolutely Continuous Invariant Measure
	C.2 Markov Property
		C.2.1 A Countable Markov Partition
			C.2.1.1 Markov Tiles
	C.3 Ergodicity
		C.3.1 Bounding Jacobian Distortion
		C.3.2 A Lebesgue Density Argument
		C.3.3 End of Proof
	References
References
Index