We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all L functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Pesk...

Stable ergodicity is dense among compact Lie group extensions of Anosov dif-feomorphisms of compact manifolds. Under the additional assumption that the base map acts on an infranilmanifold, an extension that is not stably ergodic must have a factor that has one of three special forms. A consequence is that stable ergodicity and stable ergodicity within skew products are equivalent in this case....

We discuss the long-time numerical simulation of Hamiltonian systems of ordinary differential equations. Our goal is to explain the ability of symplectic integration schemes such as Störmer-Verlet to compute accurate long-time averages for these systems in the context of molecular dynamics. This paper introduces a weakened version of ergodicity that allows us to study this problem. First, we de...

This paper studies the f-ergodicity and its exponential convergence rate for continuous-time Markov chain. Assume f is square integrable, reversible chain, it proved that of holds if only spectral gap generator positive. Moreover, equal to gap. For irreversible case, positivity remains a sufficient condition f-ergodicity. The effectiveness these results are illustrated by some typical examples.

We study the dynamics of hierarchy of piecewise maps generated by one-parameter families of trigonometric chaotic maps and one-parameter families of elliptic chaotic maps of cn and sn types, in detail. We calculate the Lyapunov exponent and Kolmogorov-Sinai entropy of the these maps with respect to control parameter. Non-ergodicity of these piecewise maps is proven analytically and investigated...

We study ergodicity of bounded, sub-additive and non-negatively homogeneous maps on finite dimensional spaces which we call upper transition operators. We show that ergodicity coincides with the necessary and sufficient condition for a generalised Perron-Frobenius theorem for upper transition operators. We show that ergodicity is equivalent with regular absorningness of the upper transition ope...

We prove weak ergodicity of the inhomogeneous Markov process generated by the generalized transition probability of Tsallis and Stariolo under power-law decay of the temperature. We thus have a mathematical foundation to conjecture convergence of simulated annealing processes with the generalized transition probability to the minimum of the cost function. An explicitly solvable example in one d...

Dynamic nuclear polarization (DNP) is to date the most effective technique to increase the nuclear polarization opening disruptive perspectives for medical applications. In a DNP setting, the interacting spin system is quasi-isolated and brought out of equilibrium by microwave irradiation. Here we show that the resulting stationary state strongly depends on the ergodicity properties of the spin...

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is bounded and the map f restricted to its closure is expanding, the property refered to as subexpanding. We first show the existence, uniqueness, conservativity ...

A weakly continuous near-action of a Polish group G on a standard Lebesgue measure space (X,μ) is whirly if for every A ⊆ X of strictly positive measure and every neighbourhood V of identity in G the set V A has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sen...