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In the present work the Calderbank-Pedersen description of four dimensional manifolds with self-dual Weyl tensor is used to obtain examples of quaternionic-kahler metrics with two commuting isometries. The eigenfunctions of the hyperbolic…
The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any $0<\beta<\alpha$, any compact metric space $X$ of Hausdorff dimension $\alpha$…
We regard the classic Thue--Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of…
We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG)…
Let $\mathbb{G}(d,n)$ be the complex Grassmannian of affine $d$-planes in $n$-space. We study the problem of characterizing the set of algebraic subvarieties of $\mathbb{G}(d,n)$ invariant under the action of the maximal torus $T$ and…
It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find…
In this paper we use the additive thermodynamic formalism to obtain new bounds of the Hausdorff and box-counting dimension of certain non conformal hyperbolic repellers defined by $C^r$, $r > 1$ piecewise expanding maps on a $d$-dimensional…
The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which gives the Hausdorff dimension for limsup sets. We investigate a related problem of estimating the Hausdorff dimension of a liminf set. Let $h>0, \tau\geq…
We construct suitable metrics for two classes of topological dynamical systems (linear maps on the torus and non-invertible expansive maps on compact spaces) in order to get a lower bound for topological entropy in terms of the resulting…
We study a self-interacting scalar $\varphi^4$ theory on the $d$-dimensional noncommutative torus. We determine, for the particular cases $d=2$ and $d=4$, the counterterms required by one-loop renormalization. We discuss higher loops in two…
We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric and study its properties, in particular completeness and local compactness, for Lorentzian manifolds and in more general…
We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance…
In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is…
In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points…
We introduce a Riemannian metric on certain hyperbolic components in the moduli space of degree $d \ge 2$ polynomials. Our metric is constructed by considering the measure-theoretic entropy of a polynomial with respect to some equilibrium…
We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized…
We investigate the dimension of the set of points in the d-torus which have the property that their orbit under rotation by some alpha hits a fixed closed target A more often than expected for all finite initial portions. An upper bound for…
Let $X \subset \mathbb{R}^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the set of…
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).
A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first…