Variations on a central limit theorem in infinite ergodic theory

In a previous paper the author proved a distributional convergence for the Birkhoff sums of functions of null average defined over a dynamical system with an infinite, invariant, ergodic measure, akin to a central limit theorem. Here we extend this result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given to continuous time systems: semi-flows, flows, and $\mathbb{Z}^{d}$-extensions of flows. The latter generalization is applied to the geodesic flow on $\mathbb{Z}^{d}$-periodic manifolds of negative sectional curvature.

Infinite-step nilsystems, independence and complexity

An∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an∞-step nilsystem is computed, showing that it is polynomial for each non-trivial open cover.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

ANALYSIS OF THE STOCHASTIC FITZHUGH–NAGUMO SYSTEM

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup Pt; further, we identify its infinitesimal generator in the space L2 (H; ν).

Bi-invariant sets and measures have integer Hausdorff dimension

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.