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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
Cybernetics : or, Control and communication in the animal and the machine
\"Cybernetics is the interdisciplinary study of controlling the flow of information in systems with feedback loops, be they biological, mechanical, cognitive, or social. This book is widely cited for laying the theoretical foundations of information theory and influencing the development of error-correcting servomechanisms, autonomous navigation, analog computing, artificial intelligence, and neuroscience\"-- Provided by publisher.
Book
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
Making better decisions using systems thinking : how to stop firefighting, deal with root causes and deliver permanent solutions
This book shows how to understand systems in order to make smarter decisions. It offers managers intervention techniques that enable them to solve the core problem instead of wasting time constantly fighting the symptoms. The most obvious part of any problem is the pain it causes. The desire to end the pain and find a solution, any solution, which will make it go away now is usually so great that it blinds managers to the underlying systemic cause of the problem. The result is that we \"solve\" the problem today and then it comes back again tomorrow or next week, again and again. We are only addressing the symptoms but never understanding the cause - like picking the flower heads off weeds but not digging them out at the roots. Schaveling and Bryan offer the insights and tools managers and leaders need to achieve a longer term and more effective approach by stepping back and analyzing the system as a whole. And at the heart of any system are human beings - notoriously short-term and pain-averse creatures who will behave in whatever way minimizes pain today even at the expense of pain tomorrow. They show how to detect the behavior patterns that have become ingrained in the organisation and which underlie complex situations so that root causes of problems can be identified. Once the system responsible for the problem is understood smarter decisions can be made to devise interventions that solve the core problem instead of wasting energy fighting the symptoms.
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
Network science in cognitive psychology
\"This volume provides an integrative review of the emerging and increasing use of network science techniques in cognitive psychology, first developed in mathematics, computer science, sociology, and physics\"-- Provided by publisher.
Book
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
Overlapping Iterated Function Systems from the Perspective of Metric Number Theory
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous
result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is
determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated
function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems
that our results apply to include those arising from Bernoulli convolutions, the
For each
Last of all, we introduce a property of an iterated function system that we call being consistently
separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous.
We include several explicit examples of consistently separated iterated function systems.
eBook