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 L ([0, 1]). In particular, there is no computable bo...

We study hierarchical properties of Sturmian words. These properties are similar to those of substitution dynamical systems. This approach allows one to carry over to Sturmian dynamical systems methods developed in the context of substitutions. For example, it allows for a proof of an ergodic type theorem for additive functions taking values in a Banach space. We then focus on establishing vari...

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s. the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–...

Gases reach equilibrium when left to themselves. Why do they behave in this way? The canonical answer to this question, originally proffered by Boltzmann, is that the systems have to be ergodic. This answer has been criticised on different grounds and is now widely regarded as flawed. In this paper we argue that some of the main arguments against Boltzmann’s answer, in particular, arguments bas...

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approac...

Pointwise and ergodic iteration-complexity results for the proximal alternating direction method of multipliers (ADMM) for any stepsize in (0, (1 + √ 5)/2) have been recently established in the literature. In addition to giving alternative proofs of these results, this paper also extends the ergodic iteration-complexity result to include the case in which the stepsize is equal to (1+ √ 5)/2. As...

A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy. A measured solenoid immersed in a smooth manifold produces a closed current (known as a generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (up to scalars) transversal measure. It is known that for any smooth manifold, any real homology class is re...

In the literature, the combination of the alternating direction method of multipliers (ADMM) with the logarithmic-quadratic proximal (LQP) regularization has been proved to be convergent and its worst-case convergence rate in the ergodic sense has been established. In this paper, we focus on a convex minimization model and consider an inexact version of the combination of the ADMM with the LQP ...

Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finitetime intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized...

Inspired by the kinetic theory of ergodic gases and chaotic billiards, we propose a simple microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, where we find exact expressions for all transport coefficients, the thermoelectric efficiency of id...