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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
Introduction to systems theory
Niklas Luhmann ranks as one of the most important sociologists and social theorists of the twentieth century. Through his many books he developed a highly original form of systems theory that has been hugely influential in a wide variety of disciplines.In Introduction to Systems Theory, Luhmann explains the key ideas of general and sociological systems theory and supplies a wealth of examples to illustrate his approach. The book offers a wide range of concepts and theorems that can be applied to politics and the economy, religion and science, art and education, organization and the family. Moreover, Luhmann's ideas address important contemporary issues in such diverse fields as cognitive science, ecology, and the study of social movements.This book provides all the necessary resources for readers to work through the foundations of systems theory - no other work by Luhmann is as clear and accessible as this. There is also much here that will be of great interest to more advanced scholars and practitioners in sociology and the social sciences.
Book
Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume
We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We
then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of
eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume
and the dynamics of their geodesic flows.
eBook
Tunneling estimates and approximate controllability for hypoelliptic equations
This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator
The first result is the tunneling estimate
The main
result is a stability estimate for solutions to the hypoelliptic wave equation
We then prove the approximate controllability of the
hypoelliptic heat equation
We also explain how the analyticity
assumption can be relaxed, and a boundary
Most results turn out to be optimal on a family of Grushin-type operators.
The main proof relies on the
general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).
eBook
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