Introduction Part I. Entropy in Ergodic Theory: 1. Shannon information and entropy 2. Dynamical entropy of a process 3. Entropy theorems in processes 4. Kolmogorov-Sinai entropy 5. The ergodic law of series Part II. Entropy in Topological Dynamics: 6. Topological entropy 7. Dynamics in dimension zero 8. The entropy structure 9. Symbolic extensions 10. A touch of smooth dynamics Part III. Entropy Theory for Operators: 11. Measure theoretic entropy of stochastic operators 12. Topological entropy of a Markov operator 13. Open problems in operator entropy Appendix A. Toolbox Appendix B. Conditional S-M-B List of symbols References Index.

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Entropy in Dynamical Systems

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/104348/1/0347-3.pdf

複素解析的特異葉層構造 (Topology of Foliations)

Foundation of the Theory of Algebraic Invariants. By G. Gurevich, Translated from the Russian by J-R. M. Radok and A. J. M. Spencer. 19 viii, 418 pp. + index. Cloth bound Dfl. 58.50 U.S. $ 16.25.

Stein–Weiss Operators and Ellipticity

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An integral formula for a Riemannian manifold with two orthogonal complementary distributions

1 Dynamical systems.- 1.1 Pseudogroups.- 1.2 First examples.- 1.3 Foliations, laminations and holonomy.- 1.4 Markov pseudogroups.- 1.5 Hyperbolic spaces and groups.- 2 Growth.- 2.1 Growth types.- 2.2 Growth in groups.- 2.3 Orbit growth for pseudogroups.- 2.4 Expansion growth.- 3 Entropy.- 3.1 Entropy of classical systems.- 3.1.1 Topological entropy of a transformation.- 3.1.2 Invariant measures.- 3.1.3 Measure-theoretic entropy.- 3.1.4 Examples.- 3.1.5 Variational principle.- 3.2 Entropy of pseudogroups.- 3.3 Geometric entropy of foliations.- 3.4 Relating various entropies.- 3.5 Examples and constructions.- 3.5.1 Pullback.- 3.5.2 Gluing.- 3.5.3 Turbulizat ion.- 3.6 Entropy and resiliency.- 4 Invariant measures.- 4.1 Basic definitions and facts.- 4.2 Transverse invariant measures and homology.- 4.3 Measures and orbit growth.- 4.4 Transverse invariant measures in codimension 1.- 4.5 Vanishing entropy and invariant measures.- 4.6 Entropy, geodesic flow and invariant measures.- 4.7 Harmonic measures.- 4.8 Patterson-Sullivan measures.- 5 Hausdorff dimension.- 5.1 Definitions and basic facts.- 5.2 Julia sets.- 5.3 Dimension in foliated manifolds.- 5.4 Dimension of a hyperbolic boundary.- 5.5 Dimension of a limit set.- 6 Varia.- 6.1 Complexity growth.- 6.2 Expansive systems.- 6.3 Pseudo-orbits and pseudoleaves.- 6.4 Generic leaves.

Dynamics of foliations, groups and pseudogroups

-1. Integral Formulae (Introduction, Preliminaries, Integral Formulae for Codimension-one foliations).-2.Variation Formulae (Introduction, Auxiliary results, Variations of extrinsic geometric quantities, Variations of general functional, Variations of particular functional, Applications and examples).-3. Extrinsic Geometric Flows (Introduction, The systems of PDE's related to EGF, Auxiliary results, Existence and uniqueness results, A solution to general case, Global existence of EGF, Variation formulae for EGF, Extrinsic geometric solitons, Applications and examples).- References.

Topics in Extrinsic Geometry of Codimension-One Foliations

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.

Conformal geometry of foliations

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Paweł Walczak

Papers

An integral formula for a Riemannian manifold with two orthogonal complementary distributions

Dynamics of foliations, groups and pseudogroups

Topics in Extrinsic Geometry of Codimension-One Foliations

Conformal geometry of foliations

On the energy of unit vector fields with isolated singularities