Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis.

Studying commuting symmetries of $p$-hyperelliptic Riemann surfaces, Bujalance and Costa found in "On symmetries of $p$-hyperelliptic Riemann surfaces" (E. Bujalalance, A.F. Costa, Math. Ann. 308(1) (1997), 31–45) upper bounds for the degree of hyperellipticity of the product of commuting $(M-q)$- and $(M-q')$-symmetries, depending on their separabilities. Here, we find necessary and sufficient conditions for an integer $p$ to be the degree of hyperellipticity of the product of two such symmetries, taking into account their separabilities. We also give some results concerning the existence and uniqueness of symmetries from which we obtain a series of important results of Natanzon concerning $M$- and $(M-1)$-symmetries.

We present a way of thinking of exponential families as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group $G^+$ of the group $G$ of all invertible elements in the algebra $\mathcal{A}$ of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class $\mathcal{D}$ of densities with respect to a given measure will happen to be representatives of equivalence classes defining a projective space in $\mathcal{A}$. The natural geometry is defined by an intrinsic group action which allows us to think of the class of positive, invertible functions $G^+$ as a homogeneous space. Also, the parallel transport in $G^+$ and $\mathcal{D}$ will be given by the original group action. Besides studying some relationships among these constructions, we examine some Riemannian geometries and provide a geometric interpretation of Pinsker's and other classical inequalities. Also we provide a geometric reinterpretation of some relationships between polynomial sequences of convolution type, probability distributions on $\mathbb{N}$ in terms of geodesics in the Banach space $\ell_1(\alpha)$.

We consider complex dynamics of a critically finite} holomorphic map from $\mathbf{P}^{k}$ to $\mathbf{P}^{k}$, which has symmetries associated with the symmetric group $S_{k+2}$ acting on $\mathbf{P}^{k}$, for each $k \ge 1$. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.

Using rather elementary and direct methods, we first recover and add on some results of Aikawa-Hirata-Lundh about the Martin boundary of a John domain. In particular we answer a question raised by these authors. Some applications are given and the case of more general second order elliptic operators is also investigated. In the last parts of the paper two potential theoretic results are shown in the framework of uniform domains or the framework of hyperbolic manifolds.

Let $\mu$ be a Borel measure on ${\mathbb R}^d$ which may be non doubling. The only condition that $\mu$ must satisfy is $\mu(B(x,r))\leq Cr^n$ for all $x\in \mathbb {R}^d$, $r>0$ and for some fixed $n$ with $0<n\leq d$. In this paper we introduce a maximal operator $N$, which coincides with the maximal Hardy-Littlewood operator if $\mu(B(x,r))\approx r^n$ for $x\in \rm{supp}(\mu)$, and we show that all $n$-dimensional Calderón-Zygmund operators are bounded on $L^p(w\,d\mu)$ if and only if $N$ is bounded on $L^p(w\,d\mu)$, for a fixed $p\in(1,\infty)$. Also, we prove that this happens if and only if some conditions of Sawyer type hold. We obtain analogous results about the weak $(p,p)$ estimates. This type of weights do not satisfy a reverse Hölder inequality, in general, but some kind of self improving property still holds. On the other hand, if $f \in \mathit{RBMO}(\mu)$ and ${\varepsilon}>0$ is small enough, then $e^{{\varepsilon} f}$ belongs to this class of weights.