Multifractal Analysis of Multiple Ergodic Averages

In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate.Comment: 5 pages, to appear in Comptes Rendus Mathematiqu

Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space $\Sigma_m=\{0,...,m-1\}^\N$ that are invariant under multiplication by integers. The results apply to the sets $\{x\in \Sigma_m: \forall\, k, \ x_k x_{2k}... x_{n k}=0\}$, where $n\ge 3$. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.Comment: 22 page

Estimates of Weyl Sums over Subsequences of Natural Numbers

In this paper we introduce the notion of pseudo ‐ergodicity to generalize Pustyl'nikov's estimates of Weyl sums to Weyl sums over subsequence of the natural numbers

On transfer operators and maps with random holes

International audienceWe study Markov interval maps with random holes. The holes are not necessarily elements of the Markov partition. Under a suitable, and physically relevant, assumption on the noise, we show that the transfer operator associated with the random open system can be reduced to a transfer operator associated with the closed deterministic system. Exploiting this fact, we show that the random open system admits a unique (meaningful) absolutely continuous conditionally stationary measure. Moreover, we prove the existence of a unique probability equilibrium measure supported on the survival set, and we study its Hausdorff dimension

MULTIFRACTAL ANALYSIS OF MULTIPLE ERGODIC AVERAGES

5 pages, to appear in Comptes Rendus Mathematique.In this paper we present a complete solution to the problem of multifractal analysis of multiple ergodic averages in the case of symbolic dynamics for functions of two variables depending on the first coordinate

THE MULTIFRACTAL SPECTRA OF V-STATISTICS

15 pages; minor revision after the referee report; to appear in Proceedings of the Conference on Fractals and Related Fields II (Porquerolles, June 2011)Let $(X, T)$ be a topological dynamical system and let $\Phi: X^r \to \mathbb{R}$ be a continuous function on the product space $X^r= X\times \cdots \times X$ ($r\ge 1$). We are interested in the limit of V-statistics taking $\Phi$ as kernel: $$\lim_{n\to \infty} n^{-r}\sum_{1\le i_1, \cdots, i_r\le n} \Phi(T^{i_1}x, \cdots, T^{i_r} x).$$ The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case ($r=1$) where the spectrum is an analytic function when $\Phi$ is H\"{o}lder continuous, the spectrum of the limit of higher order V-statistics ($r\ge 2$) may be discontinuous even for very nice kernel $\Phi$