From the Publisher:
Although it originated as a method to study general dynamical systems, symbolic dynamics is useful in coding for data storage and transmission as well as in linear algebra. Requiring only a undergraduate knowledge of linear algebra, this first general textbook includes over 500 exercises.

An Introduction to Symbolic Dynamics and Coding

The large deviation approach to statistical mechanics

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.

This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equivalent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of linear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincare maps are then derived and used to characterize the birth and the death of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurcation analysis. In particular, the analytical expressions characterizing various half-return maps associated with the Poincare map are used in a crucial way to prove the existence of a Shilnikov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and death (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in R^3 . Using this approach, we were able to prove that the chaotic dynamics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov's theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearlyequivalent circuits) of vector fields whose associated piecewise-linear differential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system's global qualitative behavior.

The double scroll family

The emerging area of control with limited data rates incorporates ideas from both control and information theory The data rate constraint introduces quantization into the feedback loop and gives the interconnected system a two-fold nature, continuous and symbolic In this paper, we review the results available in the literature on data-rate-limited control For linear systems, we show how fundamental tradeoffs between the data rate and control goals, such as stability, mean entry times, and asymptotic state norms, emerge naturally While many classical tools from both control and information theory can still be used in this context, it turns out that the deepest results necessitate a novel, integrated view of both disciplines

Feedback Control Under Data Rate Constraints: An Overview

Introduction and preliminaries Part I. Topological Dynamics: 1. Examples and basic properties 2. An application of recurrence to arithmetic progressions 3. Topological entropy 4. Interval maps 5. Hyperbolic toral automorphisms 6. Rotation numbers Part II. Measurable Dynamics: 7. Invariant measures 8. Measure theoretic entropy 9. Ergodic measures 10. Ergodic theorems 11. Mixing 12. Statistical properties Part III. Supplementary Chapters: 13. Fixed points for the annulus 14. Variational principle 15. Invariant measures for commuting transformations 16. An application of ergodic theory to arithmetic progressions.

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Dynamical Systems and Ergodic Theory

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

Thermodynamic formalism for certain nonhyperbolic maps

For countable to one transitive Markov systems we establish thermodynamic formalism for non-Holder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.

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Thermodynamic formalism for countable to one Markov systems

We give a general definition of the topological pressureP
 top
 (f, S) for continuous real valued functionsf: X→ℝ on transitive countable state Markov shifts (X, S). A variational principle holds for functions satisfying a mild distortion property. We introduce a new notion of Z-recurrent functions. Given any such functionf, we show a general method how to obtain tight sequences of invariant probability measures supported on periodic points such that a weak accumulation pointμ is an equilibrium state forf if and only if ef
 −
 dμ<∞. We discuss some conditions that ensure this integrability. As an application we obtain the Gauss measure as a weak limit of measures supported on periodic points.

Pressure and equilibrium states for countable state markov shifts

We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, ie a characterization of the radius of convergence in terms of pressure The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems The Markov approximation method then gives a product formula of zeta functions for certain weighted functions

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385798117972

Michiko Yuri

Papers

Dynamical Systems and Ergodic Theory

Thermodynamic formalism for certain nonhyperbolic maps

Thermodynamic formalism for countable to one Markov systems

Pressure and equilibrium states for countable state markov shifts

Zeta functions for certain non-hyperbolic systems and topological Markov approximations