Bounded holomorphic functions attaining their norms in the bidual

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain their norms, is dense in $\mathcal{A}_u(X)$. The result holds also for functions with values in a dual space or in a Banach space with the so-called property $(\beta)$. For this, we establish first a Lindenstrauss type theorem for continuous polynomials. We also present some counterexamples for the Bishop-Phelps theorem in the analytic and polynomial cases where our results apply.Comment: Accepted in Publ. Res. Inst. Math. Sc

A Lindenstrauss theorem for some classes of multilinear mappings

Under some natural hypotheses, we show that if a multilinear mapping belongs to some Banach multlinear ideal, then it can be approximated by multilinear mappings belonging to the same ideal whose Arens extensions simultaneously attain their norms. We also consider the class of symmetric multilinear mappings.Comment: 11 page

Mean Dimension & Jaworski-type Theorems

According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim(X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1]^{d+1})^{\mathbb{Z}},shift)$.Comment: To appear in Proceedings of the London Mathematical Societ

Entropy of convolutions on the circle

Given ergodic p-invariant measures {\mu_i} on the 1-torus T=R/Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution \muon converges to \log p. We also prove a variant of this result for joinings of full entropy on \T^\N. In conjunction with a method of Host, this yields the following. Denote \sig_q(x) = qx\pmod{1}. Then for every p-invariant ergodic \mu with positive entropy, \frac{1}{N}\sum_{n=0}^{N-1}\sig_{c_n}\mu converges weak^* to Lebesgue measure as N \goesto \infty, under a certain mild combinatorial condition on {c_k}. (For instance, the condition is satisfied if p=10 and c_k=2^k+6^k or c_k=2^{2^k}.) This extends a result of Johnson and Rudolph, who considered the sequence c_k = q^k when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorff dimension of sum sets: For any sequence {S_i} of p-invariant closed subsets of T, if \sum \dim_H(S_i) / |\log\dim_H(S_i)| = \infty, then \dim_H(S_1 + \cdots + S_n) \goesto 1.Comment: 34 pages, published versio

Rainwater-Simons-type convergence theorems for generalized convergence methods

We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.Comment: 10 pages, version 2, references added, one remark added, revised version accepted for publication in Acta et Commentationes Universitatis Tartuensis de Mathematic