We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and overview some the recent developments in mathematical Chaos.

Let P be a Markov kernel on a measurable space X and let V : X→[1,+∞). This paper provides explicit connections between the V -geometric ergodicity of P and that of finite-rank nonnegative sub-Markov kernels P̂k approximating P . A special attention is paid to obtain an efficient way to specify the convergence rate for P from that of P̂k and conversely. Furthermore, explicit bounds are obtained f...

We prove that ergodicity of the horocycle ow on a surface of constant negative curvature is equivalent to ergodicity of the associated boundary action. As a corollary we obtain ergodicity of the horocycle ow on several large classes of covering surfaces. There are two natural \geometric ows" on (the unitary tangent bundle of) an arbitrary surface of constant negative curvature: the geodesic and...

The paper is devoted to a study of the basic ergodic properties (ergodicity and conservativity) of the horocycle ow on surfaces of constant negative curvature with respect to the Liouville invariant measure. We give several criteria for ergodicity and conservativity and connect them with the classiication of the associated Fuchsian groups. Special attention is given to covering surfaces. In par...

The Nosé-Hoover dynamics is a deterministic method that is commonly used to sample the canonical Gibbs measure. This dynamics extends the physical Hamiltonian dynamics by the addition of a “thermostat” variable, that is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are con...

We prove severaìautomatic' ergodicity results for cocycles on a discrete nonsingular ergodic equivalence relation on a probability space (X; S;) with values in virtually nilpotent groups. The hypotheses required for automatic ergodicity are invari-ance or quasi-invariance of the cocycles under asymptotically central automorphisms of the equivalence relation. If the cocycles have certain recurre...

In this paper, we show that an $L_1$-2-type surface in the three-dimensional hyperbolic space $H^3subset R^4_1$ either is an open piece of a standard Riemannian product $ H^1(-sqrt{1+r^2})times S^{1}(r)$, or it has non constant mean curvature, non constant Gaussian curvature, and non constant principal curvatures.

1 Networks of binary neurons 3 1.1 Stochastic binary neurons and networks . . . . . . . . . . . . . . 3 1.1.1 Parallel dynamics: Little model . . . . . . . . . . . . . . 5 1.1.2 Sequential dynamics . . . . . . . . . . . . . . . . . . . . 5 1.2 Some properties of Markov processes . . . . . . . . . . . . . . . 6 1.2.1 Eigenvalue spectrum of T . . . . . . . . . . . . . . . . . 6 1.2.2 Ergodicity a...