Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

Ergodic theory of chaos and strange attractors

One of basic functional relationships in the Keynesian model of the economy is the liquidity preference schedule, an inverse relationship between the demand for cash balances and the rate of interest. This aggregative function must be derived from some assumptions regarding the behavior of the decision-making units of the economy, and those assumptions are the concern of this paper.

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Liquidity Preference as Behavior towards Risk

This advanced textbook introduces students to cohomology theory. No knowledge of homological algebra is assumed beyond what is normally taught in a first course in algebraic topology.

Cohomology of Groups

Preface.- 1 Smooth Manifolds.- 2 Smooth Maps.- 3 Tangent Vectors.- 4 Submersions, Immersions, and Embeddings.- 5 Submanifolds.- 6 Sard's Theorem.- 7 Lie Groups.- 8 Vector Fields.- 9 Integral Curves and Flows.- 10 Vector Bundles.- 11 The Cotangent Bundle.- 12 Tensors.- 13 Riemannian Metrics.- 14 Differential Forms.- 15 Orientations.- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem.- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.- 22 Symplectic Manifolds.- Appendix A: Review of Topology.- Appendix B: Review of Linear Algebra.- Appendix C: Review of Calculus.- Appendix D: Review of Differential Equations.- References.- Notation Index.- Subject Index

http://ndl.ethernet.edu.et/bitstream/123456789/88506/1/2012_Book_IntroductionToSmoothManifolds.pdf

Introduction to Smooth Manifolds

Metric Spaces Length Spaces Constructions Spaces of Bounded Curvature Smooth Length Structures Curvature of Riemannian Metrics Space of Metric Spaces Large-scale Geometry Spaces of Curvature Bounded Above Spaces of Curvature Bounded Below Bibliography Index.

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A Course in Metric Geometry

The description for this book, Singular Points of Complex Hypersurfaces. (AM-61), will be forthcoming.

Singular points of complex hypersurfaces

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattes map has been made more inclusive, and the ecalle-Voronin theory of parabolic points is described. The residu iteratif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.

Dynamics in one complex variable

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras and some surrounding topics. New proofs of the classical theorems are given, as well as some new results. The paper is divided into eight sections with the following titles: 1. Algebras and modules. 2. Coalgebras and comodules. 3. Algebras, coalgebras, and duality. 4. Elementary properties of Hopf algebras. 5. Universal algebras of Lie algebras. 6. Lie algebras and restricted Lie algebras. 7. Some classical theorems. 8. Morphisms of connected coalgebras into connected algebras. The first four sections are introductory in nature. Section 5 shows that, over a field of characteristic zero, the category of graded Lie algebras is isomorphic with the category of primitively generated Hopf algebras. In ? 6, a similar result is obtained in the case of characteristic p # 0, but with graded Lie algebras replaced by graded restricted Lie algebras. Section 7 studies conditions when a Hopf algebra with commutative multiplication splits either as a tensor product of algebras with a single generator or a tensor product of

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On the Structure of Hopf Algebras

Curvatures of left invariant metrics on lie groups

Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed point theorem133Proof of Sard's theorem164The degree modulo 2 of a mapping20Smooth homotopy and smooth isotopy205Oriented manifolds26The Brouwer degree276Vector fields and the Euler number327Framed cobordism the Pontryagin construction42The Hopf theorem508Exercises52AppClassifying 1-manifolds55Bibliography59Index63

John Milnor

Papers

Singular points of complex hypersurfaces

Dynamics in one complex variable

On the Structure of Hopf Algebras

Curvatures of left invariant metrics on lie groups

Topology from the differentiable viewpoint