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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.

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

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.

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.

Many developing countries are looking to scale-up what works through major systems strengthening investments. With leadership, conviction and commitment, systems thinking can facilitate and accelerate the strengthening of systems to more effectively deliver interventions to those in need and be better able to improve health in an equitable way. Systems thinking is not a panacea. Its application does not mean that resolving problems and weaknesses will come easily or naturally or without overcoming the inertia of the established way of doing things. But it will identify, with more precision, where some of the true blockages and challenges lie. It will help to: 1) explore these problems from a systems perspective; 2) show potentials of solutions that work across sub-systems; 3) promote dynamic networks of diverse stakeholders; 4) inspire learning; and 5) foster more system-wide planning, evaluation and research. And it will increase the likelihood that health system strengthening investments and interventions will be effective. The more often and more comprehensively the actors and components of the system can talk to each other from within a common framework --communicating, sharing, problem-solving - the better chance any initiative to strengthen health systems has. Real progress will undoubtedly require time, significant change, and momentum to build capacity across the system. However, the change is necessary - and needed now. This report therefore speaks to health system stewards, researchers and funders and maps out a set of strategies and activities to harness these approaches, to link them to these emerging opportunities and to assist systems thinking to become the norm in design and evaluation of interventions in health systems. But, the final message is to the funders of health system strengthening and health systems research who will need to recognize the potential in these opportunities, be prepared to take risks in investing in such innovations, and play an active role in both driving and following this agenda towards more systemic and evidence-informed health development.

We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is

This volume is a tribute to one of the founders of modern theory of dynamical systems, the late Dmitry Victorovich Anosov.It contains both original papers and surveys, written by some distinguished experts in dynamics, which are related to important themes of Anosov's work, as well as broadly interpreted further crucial developments in the theory of dynamical systems that followed Anosov's original work.Also included is an article by A. Katok that presents Anosov's scientific biography and a picture of the early development of hyperbolicity theory in its various incarnations, complete and partial, uniform and nonuniform.

We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs

This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps