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Author: David Ruelle
Publisher:
ISBN: 9781470438500
Category : Differentiable dynamical systems
Languages : en
Pages : 62
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Book Description
Consider a space M, a map f:M\to M, and a function g:M \to {\mathbb C}. The formal power series \zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\, f^m} \prod ^{m-1}_{k=0} g (f^kx) yields an example of a dynamical zeta function. Such functions have unexpected analytic properties and interesting relations to the theory of dynamical systems, statistical mechanics, and the spectral theory of certain operators (transfer operators). The first part of this monograph presents a general introduction to this subject. The second part is a detailed study of the zeta functions.
Author: David Ruelle
Publisher:
ISBN: 9781470438500
Category : Differentiable dynamical systems
Languages : en
Pages : 62
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Book Description
Consider a space M, a map f:M\to M, and a function g:M \to {\mathbb C}. The formal power series \zeta (z) = \exp \sum ^\infty _{m=1} \frac {z^m}{m} \sum _{x \in \mathrm {Fix}\, f^m} \prod ^{m-1}_{k=0} g (f^kx) yields an example of a dynamical zeta function. Such functions have unexpected analytic properties and interesting relations to the theory of dynamical systems, statistical mechanics, and the spectral theory of certain operators (transfer operators). The first part of this monograph presents a general introduction to this subject. The second part is a detailed study of the zeta functions.
Author: D. Ruelle
Publisher:
ISBN:
Category :
Languages : en
Pages : 4
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Book Description
Author: David Ruelle
Publisher: American Mathematical Soc.
ISBN: 9780821836019
Category : Mathematics
Languages : en
Pages : 74
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Book Description
With a general introduction to the subject, this title presents a detailed study of the zeta functions associated with piecewise monotone maps of the interval $ 0,1]$. In particular, it gives a proof of a generalized form of the Baladi-Keller theorem relating the poles of $\zeta (z)$ and the eigenvalues of the transfer operator.
Author: Viviane Baladi
Publisher: Springer
ISBN: 3319776614
Category : Mathematics
Languages : en
Pages : 296
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Book Description
The spectra of transfer operators associated to dynamical systems, when acting on suitable Banach spaces, contain key information about the ergodic properties of the systems. Focusing on expanding and hyperbolic maps, this book gives a self-contained account on the relation between zeroes of dynamical determinants, poles of dynamical zeta functions, and the discrete spectra of the transfer operators. In the hyperbolic case, the first key step consists in constructing a suitable Banach space of anisotropic distributions. The first part of the book is devoted to the easier case of expanding endomorphisms, showing how the (isotropic) function spaces relevant there can be studied via Paley–Littlewood decompositions, and allowing easier access to the construction of the anisotropic spaces which is performed in the second part. This is the first book describing the use of anisotropic spaces in dynamics. Aimed at researchers and graduate students, it presents results and techniques developed since the beginning of the twenty-first century.
Author: Andreas Juhl
Publisher: Birkhäuser
ISBN: 3034883404
Category : Mathematics
Languages : en
Pages : 712
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Book Description
Dynamical zeta functions are associated to dynamical systems with a countable set of periodic orbits. The dynamical zeta functions of the geodesic flow of lo cally symmetric spaces of rank one are known also as the generalized Selberg zeta functions. The present book is concerned with these zeta functions from a cohomological point of view. Originally, the Selberg zeta function appeared in the spectral theory of automorphic forms and were suggested by an analogy between Weil's explicit formula for the Riemann zeta function and Selberg's trace formula ([261]). The purpose of the cohomological theory is to understand the analytical properties of the zeta functions on the basis of suitable analogs of the Lefschetz fixed point formula in which periodic orbits of the geodesic flow take the place of fixed points. This approach is parallel to Weil's idea to analyze the zeta functions of pro jective algebraic varieties over finite fields on the basis of suitable versions of the Lefschetz fixed point formula. The Lefschetz formula formalism shows that the divisors of the rational Hassc-Wcil zeta functions are determined by the spectra of Frobenius operators on l-adic cohomology.
Author: Michel L. Lapidus
Publisher: Springer Science & Business Media
ISBN: 0387352082
Category : Mathematics
Languages : en
Pages : 472
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Book Description
Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.
Author: Alexander Fel'shtyn
Publisher: American Mathematical Soc.
ISBN: 0821820907
Category : Mathematics
Languages : en
Pages : 165
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Book Description
In the paper we study new dynamical zeta functions connected with Nielsen fixed point theory. The study of dynamical zeta functions is part of the theory of dynamical systems, but it is also intimately related to algebraic geometry, number theory, topology and statistical mechanics. The paper consists of four parts. Part I presents a brief account of the Nielsen fixed point theory. Part II deals with dynamical zeta functions connected with Nielsen fixed point theory. Part III is concerned with analog of Dold congruences for the Reidemeister and Nielsen numbers. In Part IV we explain how dynamical zeta functions give rise to the Reidemeister torsion, a very important topological invariant which has useful applications in knots theory,quantum field theory and dynamical systems.
Author: Michel Lapidus
Publisher: Springer Science & Business Media
ISBN: 1461421756
Category : Mathematics
Languages : en
Pages : 583
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Book Description
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.
Author: Mathieu Baillif
Publisher:
ISBN:
Category :
Languages : en
Pages : 21
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Book Description
Author: B. Hasselblatt
Publisher: Elsevier
ISBN: 0080533442
Category : Mathematics
Languages : en
Pages : 1231
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Book Description
Volumes 1A and 1B. These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys. The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics. Volume 1B will appear 2005.
