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4,352
result(s) for
"Hyperbolic systems"
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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
Geometric pressure for multimodal maps of the interval
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
eBook
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
The Regularity of the Linear Drift in Negatively Curved Spaces
We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is
eBook
Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data
In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.
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
Free Energy and Equilibrium States for Families of Interval Maps
We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general
hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free
energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium
states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper
semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one
varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential
tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of
correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.
eBook
Exact and Numerical Solutions of the Riemann Problem for a Conservative Model of Compressible Two-Phase Flows
In this work we study the solution of the Riemann problem for the barotropic version of the conservative symmetric hyperbolic and thermodynamically compatible (SHTC) two-phase flow model introduced in Romenski et al. (J Sci Comput 42(1):68, 2009, Quart Appl Math 65(2):259–279, 2007). All characteristic fields are carefully studied and explicit expressions are derived for the Riemann invariants and the Rankine–Hugoniot conditions. Due to the presence of multiple characteristics in the system under consideration, non-standard wave phenomena can occur. Therefore we briefly review admissibility conditions for discontinuities and then discuss possible wave interactions. In particular we will show that overlapping rarefaction waves are possible and moreover we may have shocks that lie inside a rarefaction wave. In contrast to nonconservative two phase flow models, such as the Baer–Nunziato system, we can use the advantage of the conservative form of the model under consideration. Furthermore, we show the relation between the considered conservative SHTC system and the corresponding barotropic version of the nonconservative Baer–Nunziato model. Additionally, we derive the reduced four equation Kapila system for the case of instantaneous relaxation, which is the common limit system of both, the conservative SHTC model and the non-conservative Baer–Nunziato model. Finally, we compare exact solutions of the Riemann problem with numerical results obtained for the conservative two-phase flow model under consideration, for the non-conservative Baer–Nunziato system and for the Kapila limit. The examples underline the previous analysis of the different wave phenomena, as well as differences and similarities of the three systems.
Journal Article
Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics
The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.
Journal Article