2,844 research outputs found

### 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

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