Abstract:
The conformal loop ensemble $\mathrm{CLE}_{\kappa}$ is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter $\kappa$ varies between $8/3$ and $8$; $\mathrm{CLE}_{8/3}$ is empty while $\mathrm {CLE}_8$ is a single space-filling loop. In this work, we study the geometry of the $\mathrm{CLE}$ gasket, the set of points not surrounded by any loop of the $\mathrm{CLE}$. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by $2-(8-\kappa)(3\kappa-8)/(32\kappa)$ when $4<\kappa<8$. Together with the work of Schramm-Sheffield-Wilson [Comm. Math. Phys. 288 (2009) 43-53] giving the upper bound for all $\kappa$ and the work of Nacu-Werner [J. Lond. Math. Soc. (2) 83 (2011) 789-809] giving the matching lower bound for $\kappa\le4$, this completes the determination of the $\mathrm{CLE}_{\kappa}$ gasket dimension for all values of $\kappa$ for which it is defined. The dimension agrees with the prediction of Duplantier-Saleur [Phys. Rev. Lett. 63 (1989) 2536-2537] for the FK gasket.

Abstract:
Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.

Abstract:
The Hausdorff dimension of a product XxY can be strictly greater than that of Y, even when the Hausdorff dimension of X is zero. But when X is countable, the Hausdorff dimensions of Y and XxY are the same. Diagonalizations of covers define a natural hierarchy of properties which are weaker than ``being countable'' and stronger than ``having Hausdorff dimension zero''. Fremlin asked whether it is enough for X to have the strongest property in this hierarchy (namely, being a gamma-set) in order to assure that the Hausdorff dimensions of Y and XxY are the same. We give a negative answer: Assuming CH, there exists a gamma-set of reals X and a set of reals Y with Hausdorff dimension zero, such that the Hausdorff dimension of X+Y (a Lipschitz image of XxY) is maximal, that is, 1. However, we show that for the notion of a_strong_ gamma-set the answer is positive. Some related problems remain open.

This elucidation investigates the Hausdorff dimension of the output space of multi-layer neural networks. When the factor map from the covering space of the output space to the output space has a synchronizing word, the Hausdorff dimension of the output space relates to its topological entropy. This clarifies the geometrical structure of the output space in more details.

Abstract:
We begin with a brief treatment of Hausdorff measure and Hausdorff dimension. We then explain some of the principal results in Diophantine approximation and the Hausdorff dimension of related sets, originating in the pioneering work of Vojtech Jarnik. We conclude with some applications of these results to the metrical structure of exceptional sets associated with some famous problems. It is not intended that all the recent developments be covered but they can be found in the references cited.

Abstract:
We show that at the vicinity of a generic dissipative homoclinic unfolding of a surface diffeomorphism, the Hausdorff dimension of the set of parameters for which the diffeomorphism admits infinitely many periodic sinks is at least 1/2.

Abstract:
We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting non-linearisable maps, a common hedgehog of Hausdorff dimension 1, the minimum possible.

Abstract:
David maps are generalizations of classical planar quasiconformal maps for which the dilatation is allowed to tend to infinity in a controlled fashion. In this note we examine how these maps distort Hausdorff dimension. We show \vs {enumerate} [$\bullet$] Given $\alpha$ and $\beta$ in $[0,2]$, there exists a David map $\phi:\CC \to \CC$ and a compact set $\Lambda$ such that $\Hdim \Lambda =\alpha$ and $\Hdim \phi(\Lambda)=\beta$. \vs [$\bullet$] There exists a David map $\phi:\CC \to \CC$ such that the Jordan curve $\Gamma=\phi (\Sen)$ satisfies $\Hdim \Gamma=2$.\vs {enumerate} One should contrast the first statement with the fact that quasiconformal maps preserve sets of Hausdorff dimension 0 and 2. The second statement provides an example of a Jordan curve with Hausdorff dimension 2 which is (quasi)conformally removable.

Abstract:
Penrose fractal tiling is one of the simplest generic examples for a noncommutative space. In the present work, we determine the Hausdorff dimension corresponding to a four-dimensional analogue of the so-calledPenrose Universe and show how it could be used in resolving various fundamental problems in high energy physics and cosmology.

Abstract:
In this paper, we prove that the Hausdorff dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Ya. Maltsev in 2003.