Classical And Dynamical Markov And Lagrange Spectra: Dynamical, Fractal And Arithmetic Aspects

Classical And Dynamical Markov And Lagrange Spectra: Dynamical, Fractal And Arithmetic Aspects
Author: Davi Dos Santos Lima
Publisher: World Scientific
Total Pages: 228
Release: 2020-09-18
Genre: Mathematics
ISBN: 9811225303

The book intends to give a modern presentation of the classical Markov and Lagrange spectrum, which are fundamental objects from the theory of Diophantine approximations and of their several generalizations related to Dynamical Systems and Differential Geometry. Besides presenting many classical results, the book includes several topics of recent research on the subject, connecting several fields of Mathematics — Number Theory, Dynamical Systems and Fractal Geometry.It includes topics as:

Topological and Ergodic Theory of Symbolic Dynamics

Topological and Ergodic Theory of Symbolic Dynamics
Author: Henk Bruin
Publisher: American Mathematical Society
Total Pages: 481
Release: 2022-12-21
Genre: Mathematics
ISBN: 1470472198

Symbolic dynamics is essential in the study of dynamical systems of various types and is connected to many other fields such as stochastic processes, ergodic theory, representation of numbers, information and coding, etc. This graduate text introduces symbolic dynamics from a perspective of topological dynamical systems and presents a vast variety of important examples. After introducing symbolic and topological dynamics, the core of the book consists of discussions of various subshifts of positive entropy, of zero entropy, other non-shift minimal action on the Cantor set, and a study of the ergodic properties of these systems. The author presents recent developments such as spacing shifts, square-free shifts, density shifts, $mathcal{B}$-free shifts, Bratteli-Vershik systems, enumeration scales, amorphic complexity, and a modern and complete treatment of kneading theory. Later, he provides an overview of automata and linguistic complexity (Chomsky's hierarchy). The necessary background for the book varies, but for most of it a solid knowledge of real analysis and linear algebra and first courses in probability and measure theory, metric spaces, number theory, topology, and set theory suffice. Most of the exercises have solutions in the back of the book.

The Markoff and Lagrange Spectra

The Markoff and Lagrange Spectra
Author: Thomas W. Cusick
Publisher: American Mathematical Soc.
Total Pages: 109
Release: 1989
Genre: Mathematics
ISBN: 0821815318

This book is directed at mathematicians interested in Diophantine approximation and the theory of quadratic forms and the relationship of these subjects to Markoff and Lagrange spectra. The authors have gathered and systemized numerous results from the diverse and scattered literature, much of which has appeared in rather inaccessible Russian publications. Readers will find a comprehensive overview of the theory of the Markoff and Lagrange spectra, starting with the origins of the subject in two papers of A. Markoff from 1879-80. Most of the progress since that time has occurred in the last 20 years or so, when there has been a resurgence of interest in these spectra. The authors provide an excellent exposition of these developments, in addition to presenting many proofs and correcting various errors in the literature.

Thermodynamic Formalism

Thermodynamic Formalism
Author: David Ruelle
Publisher: Cambridge University Press
Total Pages: 198
Release: 2004-11-25
Genre: Science
ISBN: 9781139455282

Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.

Non-Newtonian Calculus

Non-Newtonian Calculus
Author: Michael Grossman
Publisher: Non-Newtonian Calculus
Total Pages: 108
Release: 1972
Genre: Mathematics
ISBN: 9780912938011

The non-Newtonian calculi provide a wide variety of mathematical tools for use in science, engineering, and mathematics. They appear to have considerable potential for use as alternatives to the classical calculus of Newton and Leibniz. It may well be that these calculi can be used to define new concepts, to yield new or simpler laws, or to formulate or solve problems.

Chaos Theory Tamed

Chaos Theory Tamed
Author: Garnett Williams
Publisher: CRC Press
Total Pages: 518
Release: 1997-09-09
Genre: Mathematics
ISBN: 1482295415

This text aims to bridge the gap between non-mathematical popular treatments and the distinctly mathematical publications that non- mathematicians find so difficult to penetrate. The author provides understandable derivations or explanations of many key concepts, such as Kolmogrov-Sinai entropy, dimensions, Fourier analysis, and Lyapunov exponents.

Number Theory and Dynamical Systems

Number Theory and Dynamical Systems
Author: M. M. Dodson
Publisher: Cambridge University Press
Total Pages: 185
Release: 1989-11-09
Genre: Mathematics
ISBN: 0521369193

This volume contains selected contributions from a very successful meeting on Number Theory and Dynamical Systems held at the University of York in 1987. There are close and surprising connections between number theory and dynamical systems. One emerged last century from the study of the stability of the solar system where problems of small divisors associated with the near resonance of planetary frequencies arose. Previously the question of the stability of the solar system was answered in more general terms by the celebrated KAM theorem, in which the relationship between near resonance (and so Diophantine approximation) and stability is of central importance. Other examples of the connections involve the work of Szemeredi and Furstenberg, and Sprindzuk. As well as containing results on the relationship between number theory and dynamical systems, the book also includes some more speculative and exploratory work which should stimulate interest in different approaches to old problems.