We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesgue-measure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0, 1]). In particular, there is no computable bo...

We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. §0 Introduction Let (X,B, m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-abs...

Partially rectangular domains are compact two-dimensional Riemannian manifolds X, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family Xt of such domains parametrized by...

We prove strong convergence and asymptotic normality for the record and the weak record rate of observations of the form Yn = Xn + Tn, n ≥ 1, where (Xn)n∈Z is a stationary ergodic sequence of random variables and (Tn)n≥1 is a stochastic trend process, with stationary ergodic increments. The strong convergence result follows from the Dubins-Freedman law of large numbers and Birkhoff’s ergodic th...

Space–time block codes (STBCs) allow the diversity provided by multiple-input– multiple-output (MIMO) communication channels to be utilised, thereby increasing the capacity over ergodic fading channels or decreasing the outage probability over non-ergodic fading channels. The authors consider the deployment of orthogonal STBCs in spatially distributed mesh networks where the potentially differe...

With the growth of wireless networks, security has become a fundamental issue in wireless communications due to the broadcast nature of these networks. In this work, we consider MIMO wiretap channels in a fast fading environment, for which the overall performance is characterized by the ergodic MIMO secrecy rate. Unfortunately, the direct solution to finding ergodic secrecy rates is prohibitive...

Multiple independent molecular dynamics (MD) simulations are often carried out starting from a single protein structure or a set of conformations that do not correspond to a thermodynamic ensemble. Therefore, a significant statistical bias is usually present in the Markov state model generated by simply combining the whole MD sampling into a network whose nodes and links are clusters of snapsho...

Theorem. Suppose that G is a connected group locally isomorphic to a product of copies of SU(2) and U(1). If n > 2, then the Out(Fn)-action on Hom(Fn, G)/G is ergodic. We conjecture that Out(Fn) is ergodic on each connected component of Hom(Fn, G)/G for every compact Lie group G and n > 2. When G = U(1), then this action is just the action of GL(n,Z) on the n-torus R/Z, which is well known to b...

A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. More precise information is provided by various ergodic theorems which assert that, under certain conditions, th...