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This book consists of three volumes. The first volume contains
introductory accounts of topological dynamical systems, fi
nite-state symbolic dynamics, distance expanding maps, and ergodic
theory of metric dynamical systems acting on probability measure
spaces, including metric entropy theory of Kolmogorov and Sinai.
More advanced topics comprise infi nite ergodic theory, general
thermodynamic formalism, topological entropy and pressure.
Thermodynamic formalism of distance expanding maps and
countable-alphabet subshifts of fi nite type, graph directed Markov
systems, conformal expanding repellers, and Lasota-Yorke maps are
treated in the second volume, which also contains a chapter on
fractal geometry and its applications to conformal systems.
Multifractal analysis and real analyticity of pressure are also
covered. The third volume is devoted to the study of dynamics,
ergodic theory, thermodynamic formalism and fractal geometry of
rational functions of the Riemann sphere.
The book contains a detailed treatment of thermodynamic formalism
on general compact metrizable spaces. Topological pressure,
topological entropy, variational principle, and equilibrium states
are presented in detail. Abstract ergodic theory is also given a
significant attention. Ergodic theorems, ergodicity, and
Kolmogorov-Sinai metric entropy are fully explored. Furthermore,
the book gives the reader an opportunity to find rigorous
presentation of thermodynamic formalism for distance expanding maps
and, in particular, subshifts of finite type over a finite
alphabet. It also provides a fairly complete treatment of subshifts
of finite type over a countable alphabet. Transfer operators, Gibbs
states and equilibrium states are, in this context, introduced and
dealt with. Their relations are explored. All of this is applied to
fractal geometry centered around various versions of Bowen's
formula in the context of expanding conformal repellors, limit sets
of conformal iterated function systems and conformal graph directed
Markov systems. A unique introduction to iteration of rational
functions is given with emphasize on various phenomena caused by
rationally indifferent periodic points. Also, a fairly full account
of the classicaltheory of Shub's expanding endomorphisms is given;
it does not have a book presentation in English language
mathematical literature.
This is a one-stop introduction to the methods of ergodic theory
applied to holomorphic iteration. The authors begin with
introductory chapters presenting the necessary tools from ergodic
theory thermodynamical formalism, and then focus on recent
developments in the field of 1-dimensional holomorphic iterations
and underlying fractal sets, from the point of view of geometric
measure theory and rigidity. Detailed proofs are included.
Developed from university courses taught by the authors, this book
is ideal for graduate students. Researchers will also find it a
valuable source of reference to a large and rapidly expanding
field. It eases the reader into the subject and provides a vital
springboard for those beginning their own research. Many helpful
exercises are also included to aid understanding of the material
presented and the authors provide links to further reading and
related areas of research.
The theory of random dynamical systems originated from
stochastic
differential equations. It is intended to provide a framework
and
techniques to describe and analyze the evolution of dynamical
systems when the input and output data are known only
approximately, according to some probability distribution. The
development of this field, in both the theory and applications, has
gone in many directions. In this manuscript we introduce measurable
expanding random dynamical systems, develop the thermodynamical
formalism and establish, in particular, the exponential decay of
correlations and analyticity of the expected pressure although the
spectral gap property does not hold. This theory is then used to
investigate fractal properties of conformal random systems. We
prove a Bowen s formula and develop the multifractal formalism of
the Gibbs states. Depending on the behavior of the Birkhoff sums of
the pressure function we arrive at a natural classification of the
systems into two classes: quasi-deterministic systems, which share
many
properties of deterministic ones; and essentially random systems,
which are rather generic and never bi-Lipschitz equivalent to
deterministic systems. We show that in the essentially random case
the Hausdorff measure vanishes, which refutes a conjecture by
Bogenschutz and Ochs.Lastly, we present applications of our results
to various specific conformal random systems and positively answer
a question posed by Bruck and Buger concerning the Hausdorff
dimension of quadratic random Julia sets."
The main focus of this book is on the development of the theory of Graph Directed Markov Systems. This far-reaching generalization of the theory of conformal iterated systems can be applied in many situations, including the theory of dynamical systems. Dan Mauldin and Mariusz Urbanski include much of the necessary background material to increase the appeal of this book to graduate students as well as researchers. They also include an extensive list of references for further reading.
This text, the first of two volumes, provides a comprehensive and
self-contained introduction to a wide range of fundamental results
from ergodic theory and geometric measure theory. Topics covered
include: finite and infinite abstract ergodic theory, Young's
towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics
formalism, geometric function theory, various kinds of conformal
measures, conformal graph directed Markov systems and iterated
functions systems, semi-local dynamics of analytic functions, and
nice sets. Many examples are included, along with detailed
explanations of essential concepts and full proofs, in what is sure
to be an indispensable reference for both researchers and graduate
students.
This text, the second of two volumes, builds on the foundational
material on ergodic theory and geometric measure theory provided in
Volume I, and applies all the techniques discussed to describe the
beautiful and rich dynamics of elliptic functions. The text begins
with an introduction to topological dynamics of transcendental
meromorphic functions, before progressing to elliptic functions,
discussing at length their classical properties, measurable
dynamics and fractal geometry. The authors then look in depth at
compactly non-recurrent elliptic functions. Much of this material
is appearing for the first time in book or paper form. Both senior
and junior researchers working in ergodic theory and dynamical
systems will appreciate what is sure to be an indispensable
reference.
The focus of this book is on open conformal dynamical systems
corresponding to the escape of a point through an open Euclidean
ball. The ultimate goal is to understand the asymptotic behavior of
the escape rate as the radius of the ball tends to zero. In the
case of hyperbolic conformal systems this has been addressed by
various authors. The conformal maps considered in this book are far
more general, and the analysis correspondingly more involved. The
asymptotic existence of escape rates is proved and they are
calculated in the context of (finite or infinite) countable
alphabets, uniformly contracting conformal graph-directed Markov
systems, and in particular, conformal countable alphabet iterated
function systems. These results have direct applications to
interval maps, rational functions and meromorphic maps. Towards
this goal the authors develop, on a purely symbolic level, a theory
of singular perturbations of Perron--Frobenius (transfer) operators
associated with countable alphabet subshifts of finite type and
Hoelder continuous summable potentials. This leads to a fairly full
account of the structure of the corresponding open dynamical
systems and their associated surviving sets.
This two-volume set provides a comprehensive and self-contained
approach to the dynamics, ergodic theory, and geometry of elliptic
functions mapping the complex plane onto the Riemann sphere. Volume
I discusses many fundamental results from ergodic theory and
geometric measure theory in detail, including finite and infinite
abstract ergodic theory, Young's towers, measure-theoretic
Kolmogorov-Sinai entropy, thermodynamics formalism, geometric
function theory, various conformal measures, conformal graph
directed Markov systems and iterated functions systems, classical
theory of elliptic functions. In Volume II, all these techniques,
along with an introduction to topological dynamics of
transcendental meromorphic functions, are applied to describe the
beautiful and rich dynamics and fractal geometry of elliptic
functions. Much of this material is appearing for the first time in
book or even paper form. Both researchers and graduate students
will appreciate the detailed explanations of essential concepts and
full proofs provided in what is sure to be an indispensable
reference.
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