Search results for "ergodic theory"

Entropy, Lyapunov exponents, and rigidity of group actions

2018

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems…

Produits aléatoires d'opérateurs matrices de transfert

1988

Nous etudions le comportement asymptotique de produits aleatoires d'operateurs de Ruelle-Perron-Frobenius. Nous etendons le travail de Ruelle obtenu dans le cas homogene, au cas aleatoire.

Mode coupling approach to the ideal glass transition of molecular liquids: Linear molecules

1997

The mode coupling theory (MCT) for the ideal liquid glass transition, which was worked out for simple liquids mainly by Gotze, Sjogren, and their co-workers, is extended to a molecular liquid of linear and rigid molecules. By use of the projection formalism of Zwanzig and Mori an equation of motion is derived for the correlators S[sub lm,l[sup (prime)]m[sup (prime)]]([bold q],t) of the tensorial one-particle density rho [sub lm]([bold q],t), which contains the orientational degrees of freedom for l(greater-than)0. Application of the mode coupling approximation to the memory kernel results into a closed set of equations for S[sub lm,l[sup (prime)]m[sup (prime)]]([bold q],t), which requires t…

A generalized method for the design of ergodic sum-of-cisoids simulators for multiple uncorrelated rayleigh fading channels

2010

In this paper, we present a new method for the design of ergodic sum-of-sinusoids (SOS) simulation models for multiple uncorrelated Rayleigh fading channels. The method, which is intended for a special class of SOS models, known as sum-of-cisoids (SOC) models, can be used to generate an arbitrary number of uncorrelated Rayleigh fading waveforms with specified Doppler power spectral characteristics. This is in contrast to the SOS simulators currently available in the open literature that have been designed under the isotropic scattering assumption, which are limited to the simulation of uncorrelated channels characterized by Clarke's U-shaped Doppler power spectral density (DPSD). The excell…

Dynamics of the scenery flow and geometry of measures

2015

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a n…

Periodic measures and partially hyperbolic homoclinic classes

2019

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a tr…

Optical Hole Burning and Thermal Irreversibility of Non-Ergodic Systems: Polymers, Proteins, Glasses

1989

A spectral hole is used to probe configurational dynamics of non-ergodic systems far below the glass transition temperature

Applications de type Lasota–Yorke à trou : mesure de probabilité conditionellement invariante et mesure de probabilité invariante sur l'ensemble des …

2003

Abstract Let T :I→I be a Lasota–Yorke map on the interval I, let Y be a nontrivial sub-interval of I and g 0 :I→ R + , be a strictly positive potential which belongs to BV and admits a conformal measure m. We give constructive conditions on Y ensuring the existence of absolutely continuous (w.r.t. m) conditionally invariant probability measures to nonabsorption in Y. These conditions imply also existence of an invariant probability measure on the set X∞ of points which never fall into Y. Our conditions allow rather “large” holes.

Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

2018

This paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonallyinvariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $%L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $%\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an …

Uniformly ergodic A-contractions on Hilbert spaces

2009