/pdf/introduction-to-the-modern-theory-of-dynamical-systems-1eced70mum.pdf

Introduction to the Modern Theory of Dynamical Systems

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Trees and Networks

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

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First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

The purpose of this study was to evaluate patient factors that might contribute to a poor result after total knee arthroplasty (TKA). Seventy-one knees (6.9%) of 1024 primary TKAs were identified at 1 year follow-up as having a poor result because of either stiffness or pain. Radiographs demonstrated well-fixed and aligned implants. This group was compared with a matched control group of 148 nonpainful or stiff TKAs, with similar range of motion preoperatively. Logistic regression analysis was performed to compare age, sex, body mass index, comorbidities, previous surgeries, preoperative narcotic use, tobacco or alcohol use, work status, insurance status, and any history of depression. Factors that were significantly associated with a stiff or painful outcome included female sex, higher body mass index, previous knee surgery, patients on disability, diabetes mellitus, pulmonary disease, and depression.

Looks good but feels bad: factors that contribute to poor results after total knee arthroplasty.

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of nitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a nitely generated group. This paper is the rst in a sequence of papers proving results announced in our 2007 article \Quasi-isometries and rigidity of solvable groups." In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classication of lamplighter groups. The proofs of those results are completed in \Coarse dierentiation of quasi-isometries II; Rigidity for lattices in Sol and Lamplighter groups." The method used here is based on the idea of coarse dierentiation introduced in our 2007 article.

http://annals.math.princeton.edu/wp-content/uploads/annals-v176-n1-p03-p.pdf

Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs

This paper is a survey on the {\em Zimmer program}. In it's broadest form, this program seeks an understanding of actions of large groups on compact manifolds. The goals of this survey are $(1)$ to put in context the original questions and conjectures of Zimmer and Gromov that motivated the program, $(2)$ to indicate the current state of the art on as many of these conjectures and questions as possible and $(3)$ to indicate a wide variety of open problems and directions of research.

Groups acting on manifolds: around the Zimmer program

In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasi-isometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to $R \ltimes R^n$ where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasi-isometries of the solvable Lie group. Our classification of self quasi-isometries for $R \ltimes \R^n$ proves a conjecture made by Farb and Mosher in [FM4]. 
Our techniques for studying quasi-isometries extend to some other classes of groups and spaces. In particular, we characterize groups quasi-isometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain Diestel-Leader graphs are not quasi-isometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW],[Wo1]. We also prove that certain non-unimodular, non-hyperbolic solvable Lie groups are not quasi-isometric to finitely generated groups. 
The results in this paper are contributions to Gromov's program for classifying finitely generated groups up to quasi-isometry [Gr2]. We introduce a new technique for studying quasi-isometries, which we refer to as "coarse differentiation".

https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2007/0003/0004/PAMQ-2007-0003-0004-a003.pdf

Quasi-isometries and Rigidity of Solvable Groups

Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple Lie groups and their lattices in [FM2].

/pdf/almost-isometric-actions-property-t-and-local-rigidity-4lrpdg1opk.pdf

David Fisher

Papers

Looks good but feels bad: factors that contribute to poor results after total knee arthroplasty.

Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs

Groups acting on manifolds: around the Zimmer program

Quasi-isometries and Rigidity of Solvable Groups

Almost isometric actions, property (T), and local rigidity