This volume contains a collection of articles from the special program on algebraic and topologic... more This volume contains a collection of articles from the special program on algebraic and topological dynamics and a workshop on dynamical systems held at the Max-Planck Institute (Bonn, Germany). It reflects the extraordinary vitality of dynamical systems in its interaction with a broad range of mathematical subjects.
Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy and chaos.
The book is suitable for graduate students and researchers interested in dynamical systems.
This text is a rigorous introduction to ergodic theory, developing the machinery of conditional m... more This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.
Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.
Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
An Introduction to Number Theory provides an introduction to the main streams of number theory. S... more An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject.
In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.
A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.
Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.
Recurrence sequences are of great intrinsic interest and have been a central part of number theor... more Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.
The main theme of the book is the theory of heights as they appear in various guises. This includ... more The main theme of the book is the theory of heights as they appear in various guises. This includes a large body of results on Mahler's measure of the height of a polynomial of which topic there is no book available. The genesis of the measure in a paper by Lehmer is looked at, which is extremely well-timed due to the revival of interest following the work of Boyd and Deninger on special values of Mahler's measure. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations. A large chunk of the book has been devoted to the elliptic Mahler's measure. Special calculation have been included and will be self-contained. One of the most important results about Mahler's measure is that it is the entropy associated to a dynamical system. The authors devote space to discussing this and to giving some convincing and original examples to explain this phenomenon.
These notes describe several situations in dynamics where the notion of valuations on rings provi... more These notes describe several situations in dynamics where the notion of valuations on rings provides a simple language in which to describe and exploit (analogues of) hyperbolicity. There are two advantages to this approach that go a little beyond simply providing a convenient language. The first of these is that, roughly speaking, a map that is not hyperbolic with respect to a familiar distance (for example, Euclidean distance on Rn) may nonetheless look hyperbolic with respect to a different distance measure.
Abstract. Let be a Zd {action by homeomorphisms of a compact metric space (X;) xing some point y.... more Abstract. Let be a Zd {action by homeomorphisms of a compact metric space (X;) xing some point y. A point x in X is said to be homoclinic to y if (nx; y)! 0 as n leaves nite sets in Zd. Using ideas from Boyle and Lind's study of expansive subdynamics, we show that homoclinic behaviour for an expansive action is constant in expansive cones. Examples show that the homoclinic behaviour undergoes transitions as the hyperplanes separating the expansive cones are crossed.
Abstract: We study the behaviour of the dynamical zeta function and the orbit Dirichlet series fo... more Abstract: We study the behaviour of the dynamical zeta function and the orbit Dirichlet series for products of maps. The behaviour under products of the radius of convergence for the zeta function, and the abscissa of convergence for the orbit Dirichlet series, are discussed. The orbit Dirichlet series of the cartesian cube of a map with one orbit of each length is shown to have a natural boundary.
Abstract: A system of transformations is associated to a rational point on an elliptic curve. The... more Abstract: A system of transformations is associated to a rational point on an elliptic curve. The sequence entropy is connected to the canonical height, and in some cases there is a canonically defined quotient system whose entropy is the canonical height and for which the fibre entropy is accounted for by local heights at primes of bad reduction. The proofs use transcendence theory and a strong form of Siegel's theorem. We go on to extend these ideas to the morphic heights of Call and Goldstine.
Abstract. A general framework for investigating topological actions of Zd on compact metric space... more Abstract. A general framework for investigating topological actions of Zd on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lowerdimensional subspaces of Rd. Here we completely describe this expansive behavior for the class of algebraic Zd-actions given by commuting automorphisms of compact abelian groups.
Abstract: An existing dialogue between number theory and dynamical systems is advanced. A combina... more Abstract: An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the $ p $-parts of an integer sequence themselves count periodic points are studied.
Abstract Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate t... more Abstract Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L (f, g, s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's 'false Tate curve'congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.
These notes describe two constructions of measure-preserving transformations, one probabilistic a... more These notes describe two constructions of measure-preserving transformations, one probabilistic and one geometric. The notes are a supplement to the book [5]. The material is all standard, and in particular may largely be found in the monograph of Cornfeld, Fomin and Sinaı [4]. It is assembled here to provide a convenient source.
i= 1 µ (Ai) for all measurable sets A1,..., Ak. Update: Considerable progress has been made on th... more i= 1 µ (Ai) for all measurable sets A1,..., Ak. Update: Considerable progress has been made on this circle of problems. Masser [17] proved a conjecture of Schmidt [20] by showing that the order of mixing for an algebraic Zd by automorphisms of a zerodimensional group as detected by studying mixing shapes coincides with the real order of mixing. It remains a considerable problem to actually compute either for non-trivial examples.
Automorphisms of compact metric groups provide a simple family of dynamical systems with addition... more Automorphisms of compact metric groups provide a simple family of dynamical systems with additional structure, rendering them particularly amenable to detailed analysis. On the other hand, they are rigid in the sense that they cannot be smoothly perturbed, and for a fixed compact metric group the group of automorphisms is itself countable and discrete. Thus it is not clear which, if any, of their dynamical properties can vary continuously.
ABSTRACT This paper draws preliminary attention to the recent discovery of evidence for early Hol... more ABSTRACT This paper draws preliminary attention to the recent discovery of evidence for early Holocene human presence at Howburn, near Biggar, in the interior of southern Scotland. Implement types invite comparison with so-called Star Carr type Early Mesolithic assemblages of the 10 th millennium BP, or possibly with even earlier technological traditions.
Die Theorie der dynamischen Systeme liefert eine wirkungsvolle Methode um unterschiedliche Proble... more Die Theorie der dynamischen Systeme liefert eine wirkungsvolle Methode um unterschiedliche Probleme von Planetenbahnen bis hin zur klassischen Zahlentheorie zu untersuchen. Indem Einsiedler, Katok und Lindenstrauss in ihren neuesten Arbeiten Methoden aus der Theorie der dynamische Systemen benutzen, erzielen sie wichtige Fortschritte zur Lösung einer klassischen Vermutung der Zahlentheorie.
Number theory is a branch of pure mathematics concerned with the properties of numbers in general... more Number theory is a branch of pure mathematics concerned with the properties of numbers in general, and integers in particular.
This volume contains a collection of articles from the special program on algebraic and topologic... more This volume contains a collection of articles from the special program on algebraic and topological dynamics and a workshop on dynamical systems held at the Max-Planck Institute (Bonn, Germany). It reflects the extraordinary vitality of dynamical systems in its interaction with a broad range of mathematical subjects.
Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy and chaos.
The book is suitable for graduate students and researchers interested in dynamical systems.
This text is a rigorous introduction to ergodic theory, developing the machinery of conditional m... more This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.
Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.
Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
An Introduction to Number Theory provides an introduction to the main streams of number theory. S... more An Introduction to Number Theory provides an introduction to the main streams of number theory. Starting with the unique factorization property of the integers, the theme of factorization is revisited several times throughout the book to illustrate how the ideas handed down from Euclid continue to reverberate through the subject.
In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.
A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.
Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.
Recurrence sequences are of great intrinsic interest and have been a central part of number theor... more Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.
The main theme of the book is the theory of heights as they appear in various guises. This includ... more The main theme of the book is the theory of heights as they appear in various guises. This includes a large body of results on Mahler's measure of the height of a polynomial of which topic there is no book available. The genesis of the measure in a paper by Lehmer is looked at, which is extremely well-timed due to the revival of interest following the work of Boyd and Deninger on special values of Mahler's measure. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations. A large chunk of the book has been devoted to the elliptic Mahler's measure. Special calculation have been included and will be self-contained. One of the most important results about Mahler's measure is that it is the entropy associated to a dynamical system. The authors devote space to discussing this and to giving some convincing and original examples to explain this phenomenon.
These notes describe several situations in dynamics where the notion of valuations on rings provi... more These notes describe several situations in dynamics where the notion of valuations on rings provides a simple language in which to describe and exploit (analogues of) hyperbolicity. There are two advantages to this approach that go a little beyond simply providing a convenient language. The first of these is that, roughly speaking, a map that is not hyperbolic with respect to a familiar distance (for example, Euclidean distance on Rn) may nonetheless look hyperbolic with respect to a different distance measure.
Abstract. Let be a Zd {action by homeomorphisms of a compact metric space (X;) xing some point y.... more Abstract. Let be a Zd {action by homeomorphisms of a compact metric space (X;) xing some point y. A point x in X is said to be homoclinic to y if (nx; y)! 0 as n leaves nite sets in Zd. Using ideas from Boyle and Lind's study of expansive subdynamics, we show that homoclinic behaviour for an expansive action is constant in expansive cones. Examples show that the homoclinic behaviour undergoes transitions as the hyperplanes separating the expansive cones are crossed.
Abstract: We study the behaviour of the dynamical zeta function and the orbit Dirichlet series fo... more Abstract: We study the behaviour of the dynamical zeta function and the orbit Dirichlet series for products of maps. The behaviour under products of the radius of convergence for the zeta function, and the abscissa of convergence for the orbit Dirichlet series, are discussed. The orbit Dirichlet series of the cartesian cube of a map with one orbit of each length is shown to have a natural boundary.
Abstract: A system of transformations is associated to a rational point on an elliptic curve. The... more Abstract: A system of transformations is associated to a rational point on an elliptic curve. The sequence entropy is connected to the canonical height, and in some cases there is a canonically defined quotient system whose entropy is the canonical height and for which the fibre entropy is accounted for by local heights at primes of bad reduction. The proofs use transcendence theory and a strong form of Siegel's theorem. We go on to extend these ideas to the morphic heights of Call and Goldstine.
Abstract. A general framework for investigating topological actions of Zd on compact metric space... more Abstract. A general framework for investigating topological actions of Zd on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lowerdimensional subspaces of Rd. Here we completely describe this expansive behavior for the class of algebraic Zd-actions given by commuting automorphisms of compact abelian groups.
Abstract: An existing dialogue between number theory and dynamical systems is advanced. A combina... more Abstract: An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the $ p $-parts of an integer sequence themselves count periodic points are studied.
Abstract Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate t... more Abstract Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L (f, g, s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's 'false Tate curve'congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.
These notes describe two constructions of measure-preserving transformations, one probabilistic a... more These notes describe two constructions of measure-preserving transformations, one probabilistic and one geometric. The notes are a supplement to the book [5]. The material is all standard, and in particular may largely be found in the monograph of Cornfeld, Fomin and Sinaı [4]. It is assembled here to provide a convenient source.
i= 1 µ (Ai) for all measurable sets A1,..., Ak. Update: Considerable progress has been made on th... more i= 1 µ (Ai) for all measurable sets A1,..., Ak. Update: Considerable progress has been made on this circle of problems. Masser [17] proved a conjecture of Schmidt [20] by showing that the order of mixing for an algebraic Zd by automorphisms of a zerodimensional group as detected by studying mixing shapes coincides with the real order of mixing. It remains a considerable problem to actually compute either for non-trivial examples.
Automorphisms of compact metric groups provide a simple family of dynamical systems with addition... more Automorphisms of compact metric groups provide a simple family of dynamical systems with additional structure, rendering them particularly amenable to detailed analysis. On the other hand, they are rigid in the sense that they cannot be smoothly perturbed, and for a fixed compact metric group the group of automorphisms is itself countable and discrete. Thus it is not clear which, if any, of their dynamical properties can vary continuously.
ABSTRACT This paper draws preliminary attention to the recent discovery of evidence for early Hol... more ABSTRACT This paper draws preliminary attention to the recent discovery of evidence for early Holocene human presence at Howburn, near Biggar, in the interior of southern Scotland. Implement types invite comparison with so-called Star Carr type Early Mesolithic assemblages of the 10 th millennium BP, or possibly with even earlier technological traditions.
Die Theorie der dynamischen Systeme liefert eine wirkungsvolle Methode um unterschiedliche Proble... more Die Theorie der dynamischen Systeme liefert eine wirkungsvolle Methode um unterschiedliche Probleme von Planetenbahnen bis hin zur klassischen Zahlentheorie zu untersuchen. Indem Einsiedler, Katok und Lindenstrauss in ihren neuesten Arbeiten Methoden aus der Theorie der dynamische Systemen benutzen, erzielen sie wichtige Fortschritte zur Lösung einer klassischen Vermutung der Zahlentheorie.
Number theory is a branch of pure mathematics concerned with the properties of numbers in general... more Number theory is a branch of pure mathematics concerned with the properties of numbers in general, and integers in particular.
Word 2000, Word XP). TeX2Word needs Design Science MathType installed (version 4 or later); Word2... more Word 2000, Word XP). TeX2Word needs Design Science MathType installed (version 4 or later); Word2TeX does not and is able to support all versions of Equation Editor and MathType. Word2TeX provides the option to save a Word document as TeX via the usual 'Save as…'command. TeX2Word provides the option to 'Open'a TeX document from Word, converting it into Word format. Despite the large number of optional parameters, the software is very easy and intuitive to use.
For mixing Z^d-actions generated by commuting automorphisms of a compact abelian group, we invest... more For mixing Z^d-actions generated by commuting automorphisms of a compact abelian group, we investigate the directional uniformity of the rate of periodic point distribution and mixing. When each of these automorphisms has finite entropy, it is shown that directional mixing and directional convergence of the uniform measure supported on periodic points to Haar measure occurs at a uniform rate independent of the direction.
Problems related to the existence of integral and rational points on cubic curves date back at le... more Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a nonsingular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. Nonetheless, a conjecture of Hall suggests a bound on the size of integral solutions in terms of the coefficients of the defining equation. It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense. We describe this conjecture as an illustration of an underlying motif—repulsion—in the theory of Diophantine equations.
We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is wel... more We examine the diffraction properties of lattice dynamical systems of algebraic origin. It is well-known that diverse dynamical properties occur within this class. These include different orders of mixing (or higher-order correlations), the presence or absence of measure rigidity (restrictions on the set of possible shift-invariant ergodic measures to being those of algebraic origin), and different entropy ranks (which may be viewed as the maximal spatial dimension in which the system resembles an i.i.d. process). Despite these differences, it is shown that the resulting diffraction spectra are essentially indistinguishable, thus raising further difficulties for the inverse problem of structure determination from diffraction spectra. Some of them may be resolved on the level of higher-order correlation functions, which we also briefly compare.
Example 2: Let f (x) = x 6 − 2x 5 − 5x 4 − 3x 3 − 5x 2 − 2x + 1 and write Q(a), where a is a comp... more Example 2: Let f (x) = x 6 − 2x 5 − 5x 4 − 3x 3 − 5x 2 − 2x + 1 and write Q(a), where a is a complex root of f . Then a and b = 2a 5 − 6a 4 − 3a 3 − 6a 2 − 6a are fundamental units for the ring of integers in Q(a).
Mathematics - pure mathematics particularly - is impossible to talk about in detail: it is overpo... more Mathematics - pure mathematics particularly - is impossible to talk about in detail: it is overpoweringly technical, built on a large edifice of knowledge. Each word or symbol can be laden with meaning, and this intense compression of ideas is necessary to get anywhere.
Instead I will try to trace an idea (`measure rigidity') and use this to illustrate some themes (historical scope, incremental nature of developments, mathematics as a collaborative endeavour). The target is a recent result by Einsiedler, Katok and Lindenstrauss that uses ergodic theory to make progress with a problem in number theory.
We show that many algebraic actions of higher-rank abelian groups on zero-dimensional compact abe... more We show that many algebraic actions of higher-rank abelian groups on zero-dimensional compact abelian groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov-Rokhlin formula for half-space entropies.
Let X_1, X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition.... more Let X_1, X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map from X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has finite topological entropy.
Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems w... more Analogues of the prime number theorem and Merten's theorem are well-known for dynamical systems with hyperbolic behaviour. In this paper a 3-adic extension of the circle doubling map is studied. The map has a 3-adic eigendirection in which it behaves like an isometry, and the loss of hyperbolicity leads to weaker asymptotic results on orbit counting than those obtained for hyperbolic maps.
A framework for understanding the geometry of continuous actions of Z^d was developed by Boyle an... more A framework for understanding the geometry of continuous actions of Z^d was developed by Boyle and Lind using the notion of expansive behaviour along lower-dimensional subspaces. For algebraic Z^d-actions of entropy rank one, the expansive subdynamics are readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank-one action determine the expansive subdynamics. Moreover, the finer structure of the non-expansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.
There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical syste... more There are well-known analogs of the prime number theorem and Mertens' theorem for dynamical systems with hyperbolic behaviour. Here we consider the same question for the simplest non-hyperbolic algebraic systems. The asymptotic behaviour of the orbit-counting function is governed by a rotation on an associated compact group, and in simple examples we exhibit uncountably many different asymptotic growth rates for the orbit-counting function. Mertens' Theorem also holds in this setting, with an explicit rational leading coefficient obtained from arithmetic properties of the non-hyperbolic eigendirections.
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Books by Thomas Ward
Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy and chaos.
The book is suitable for graduate students and researchers interested in dynamical systems.
Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.
Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.
A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.
Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.
Papers by Thomas Ward
Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy and chaos.
The book is suitable for graduate students and researchers interested in dynamical systems.
Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.
Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
In particular, the book shows how the Fundamental Theorem of Arithmetic, handed down from antiquity, informs much of the teaching of modern number theory. The result is that number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.
A number of different approaches to number theory are presented, and the different streams in the book are brought together in a chapter that describes the class number formula for quadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behind modern computational number theory and its applications in cryptography.
Written for graduate and advanced undergraduate students of mathematics, this text will also appeal to students in cognate subjects who wish to learn some of the big ideas in number theory.
Instead I will try to trace an idea (`measure rigidity') and use this to illustrate some themes (historical scope, incremental nature of developments, mathematics as a collaborative endeavour). The target is a recent result by Einsiedler, Katok and Lindenstrauss that uses ergodic theory to make progress with a problem in number theory.