New Trends in Lyapunov Exponents: NTLE, Lisbon, Portugal, February 7–11, 2022 (CIM Series in Mathematical Sciences)

Preface
Contents
Lyapunov Exponents for Linear Homogeneous Differential Equations
	1 Introduction
		1.1 Linear Differential Systems
		1.2 Kinetic Cocycles
		1.3 The Harmonic Oscillator
		1.4 The Main Goal
	2 Kinetic Linear Cocycles
		2.1 Linear Cocycles
		2.2 Kinetic Linear Cocycles
		2.3 Topologies
			2.3.1 The Lp Topology
			2.3.2 Uniform Topologies
			2.3.3 The Cr Topology
		2.4 Lyapunov Exponents
		2.5 The Search for Positive Lyapunov Exponents
	3 The Lp Case
		3.1 Towards Zero Lyapunov Exponents
		3.2 Towards Non-zero Lyapunov Exponents
	4 The C0 Case
		4.1 Hyperbolicity
		4.2 A Mechanism for Obtaining Zero Lyapunov Exponents
	5 The Cr (r>0) Case
	6 What About Third Order Linear Homogeneous Differential Equations?
		6.1 Jerk Equations
		6.2 Partial Hyperbolicity
		6.3 Removing Zero Lyapunov Exponents on a Certain Jerk Equation
	References
An Invitation to SL2(R) Cocycles Over Random Dynamics
	1 Continuous Random Cocycles
		1.1 Continuous Random Cocycles and Lyapunov Exponent
		1.2 Uniform Hyperbolicity
		1.3 Regularity for Continuous Cocycles
		1.4 Continuous Cocycles with Positive Lyapunov Exponent
	2 Locally Constant Cocycles
		2.1 Random Product of Matrices
		2.2 Stationary Measures and Criteria for Positivity of the LE
		2.3 Regularity for Locally Constant
		2.4 Sharp Modulus of Continuity
		2.5 Regularity and Dimension of the Stationary Measures
	3 Holder Random Cocycles
		3.1 Fiber Bunched Cocycles
		3.2 Holonomies
		3.3 Su-States
		3.4 Typical Fiber Bunched Cocycles
		3.5 Continuity of the Lyapunov Exponent for Fiber Bunched Cocycles
		3.6 Modulus of Continuity of LE for Fiber Bunched Cocycles
	4 Conclusion and General Models
		4.1 More General Dynamics
	References
Randomness Versus Quasi-Periodicity
	1 Introduction
	2 Mixed Random-Quasiperiodic Cocycles
	3 Positivity of the Lyapunov Exponent
	4 Interlude: An Abstract LDT Theorem
	5 Regularity of the Lyapunov Exponent
	6 Stability of the Lyapunov Exponent
	References
Hyperbolicity or Zero Lyapunov Exponents for C2-Hamiltonians
	1 Introduction
	2 Basic Definitions
	3 Sketch of the Proof
	References
Generalized Lyapunov Exponents and Aspects of the Theory of Deep Learning
	1 Introduction
	2 Deep Learning
	3 Elements of a Metric Functional Analysis
	4 Multiplicative Ergodic Theorems for Linear Operators
	5 Diffeomorphisms
	6 Applications of Metrics and Ergodic Theorems in Deep Learning
	References
On the Multifractal Formalism of Lyapunov Exponents: A Survey of Recent Results
	1 Introduction
	2 Notation and Basic Definitions
		2.1 Symbolic Dynamics
		2.2 Multilinear Algebra
		2.3 Legendre Transform
		2.4 Topological Entropy
	3 Multifractal Analysis of Birkhoff Averages
	4 Multifractal Analysis of Lyapunov Exponents
		4.1 Subadditive Thermodynamic Formalism
		4.2 An Example of Subadditive Potentials: Matrix Cocycles
		4.3 Multifractal Analysis of the Top Lyapunov Exponent
	5 Multifractal Analysis of All Lyapunov Exponents
		5.1 Multifractal Analysis of Vectors of Lyapunov Exponents for Dominated Matrix Cocycles
			5.1.1 Domination
			5.1.2 Cone-Criterion
		5.2 Multifractal Formalism of Vectors of Lyapunov Exponents for Generic Matrix Cocycles
			5.2.1 Typical Coycles
	References
The Continuity Problem of Lyapunov Exponents
	1 Introduction
	2 Preliminaries
		2.1 Linear Cocycles
		2.2 Lyapunov Exponents
		2.3 Continuity Problem
	3 Random Cocycles
		3.1 Continuity
		3.2 Hölder Continuity
		3.3 Analyticity
	4 Mixed Random-Quasiperiodic Cocycles
	References
Some Questions and Remarks on Lyapunov Irregular Behavior for Linear Cocycles
	1 Introduction
	2 Statement of the Main Results
	3 Preliminaries
		3.1 Hyperbolicity, Parabolicity and Elipticity
		3.2 Oseledets\' Theorem
		3.3 GL(2,R)-Valued Cocycles
		3.4 Lyapunov Irregular Points
	4 Proofs
		4.1 Proof of Theorem 2.1
		4.2 Proof of Corollary 2.1
	References