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"Dynamics"
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Symbolic Extensions of Amenable Group Actions and the Comparison Property
In topological dynamics, the
Of course, the statement is preceded by the
presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of
eBook
Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact
Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general
class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed
in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still
restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates
whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have
measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence
of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous
potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations
for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of
the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit
Theorem with an exponentially small error.
eBook
Conformal Graph Directed Markov Systems on Carnot Groups
We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped
with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit
set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples
of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include
the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the
non-real classical rank one hyperbolic spaces.
eBook
Elements of fluid dynamics
It provides an introduction to fluid dynamics in an accessible way. This title follows the idea that both the generation mechanisms and the main features of the fluid dynamic loads can be satisfactorily understood only after the equations of fluid motion and all their physical and mathematical implications have been thoroughly assimilated.
Book
Thermodynamics
This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.
eBook
Rigidity phenomena and the statistical properties of group actions on $\\text {CAT}(0)$ cube complexes
We compare the marked length spectra of some pairs of proper and cocompact cubical actions of a nonvirtually cyclic group on
$\\mathrm {CAT}(0)$
cube complexes. The cubulations are required to be virtually co-special, have the same sets of convex-cocompact subgroups, and admit a contracting element. There are many groups for which these conditions are always fulfilled for any pair of cubulations, including nonelementary cubulable hyperbolic groups, many cubulable relatively hyperbolic groups, and many right-angled Artin and Coxeter groups. For these pairs of cubulations, we study the Manhattan curve associated to their combinatorial metrics. We prove that this curve is analytic and convex, and a straight line if and only if the marked length spectra are homothetic. The same result holds if we consider invariant combinatorial metrics in which the lengths of the edges are not necessarily one. In addition, for their standard combinatorial metrics, we prove a large deviations theorem with shrinking intervals for their marked length spectra. We deduce the same result for pairs of word metrics on hyperbolic groups. The main tool is the construction of a finite-state automaton that simultaneously encodes the marked length spectra of both cubulations in a coherent way, in analogy with results about (bi)combable functions on hyperbolic groups by Calegari-Fujiwara [14]. The existence of this automaton allows us to apply the machinery of thermodynamic formalism for suspension flows over subshifts of finite type, from which we deduce our results.
Journal Article