We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.

La marche aléatoire (ou marche au hasard) est un objet fondamental de la théorie des probabilités. Un des problèmes les plus intéressants pour la marche aléatoire (ainsi que pour le mouvement brownien, son analogue dans un contexte continu) est de savoir comment elle recouvre des ensembles où se trouvent les points qui sont souvent (ou au contraire, rarement) visités, et combien il y a de tels points. Les travaux de Dembo, Peres, Rosen et Zeitouni permettent de résoudre plusieurs conjectures importantes...

Dans ce travail, nous construisons explicitement deux isomorphismes métriques partout continus. L’un entre le système dynamique symbolique associé à la substitution $\sigma :0\mapsto 01,1\mapsto 02,2\mapsto 0$ et une rotation sur le tore ${𝕋}^2$; l’autre, entre le système adique stationnaire [33] associé à la matrice de la substitution et la même rotation. Pour cela, nous étudions les propriétés arithmétiques de la frontière d’un ensemble compact de $ℂ$ appelé “fractal de Rauzy”. Les constructions se généralisent aux substitutions de la forme ${\sigma }_k:0\mapsto 01,1\mapsto 02,\cdots k-1\mapsto 0k,k\mapsto 0$...

We consider an inhomogeneous measure μ with the inhomogeneous part a self-similar measure ν, and show that for a given r ∈ (0,∞) the lower and the upper quantization dimensions of order r of μ are bounded below by the quantization dimension $D_r\left(\nu \right)$ of ν and bounded above by a unique number ${\kappa }_r\in (0,\infty )$, related to the temperature function of the thermodynamic formalism that arises in the multifractal analysis of μ.

The quantization dimension function for the image measure of a shift-invariant ergodic measure with bounded distortion on a self-conformal set is determined, and its relationship to the temperature function of the thermodynamic formalism arising in multifractal analysis is established.

We estimate the spreading of the solution of the Schrödinger equation asymptotically in time, in term of the fractal properties of the associated spectral measures. For this, we exhibit a lower bound for the moments of order $p$ at time $T$ for the state $\psi$ defined by $[\frac{1}{T}{\int }_0^T\parallel |X{|}^{p/2}{\mathrm{e}}^{-itH}\psi {\parallel }^2\mathrm{d}t]$. We show that this lower bound can be expressed in term of the generalized Rényi dimension of the spectral measure ${\mu }_{\psi }$ associated to the hamiltonian $H$ and the state $\psi$. We especially concentrate on continuous models.

The Sierpinski gasket and other self-similar fractal subsets of Rd, d ≥ 2, can be mapped by quasiconformal self-maps of Rd onto sets of Hausdorff dimension arbitrarily close to one. In R2 we construct explicit mappings. In Rd, d ≥ 3, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we present geometric conditions ensuring that (i) isomorphic systems have quasisymmetrically...

In this paper, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc.28 (1974) 174–192] relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

In this paper, we extend the results of Orey and Taylor [S. Orey and S.J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. 28 (1974) 174–192] relative to random fractals generated by oscillations of Wiener processes to a multivariate framework. We consider a setup where Gaussian processes are indexed by classes of functions.

It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f(kx-n)+{\sum }_{n\in \mathbb{Z}}{c}_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in \mathbb{Z}}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f(kx-y)dy$ has also various interesting applications....

We first provide an approach to the conjecture of Bierstone-Milman-Pawłucki on Whitney’s problem on $C^d$ extendability of functions. For example, the conjecture is affirmative for classical fractal sets. Next, we give a sharpened form of Spallek’s theorem on flatness.

By iterating the Bolyai-Rényi transformation $T\left(x\right)={(x+1)}^2(mod1)$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$x=-1+\sqrt{x_1+\sqrt{x_2+\cdots +\sqrt{x_n+\cdots }}}$$ with digits $x_n\in \{0,1,2\}$ for all $n\in \mathbb{N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_n(x,i)$ be the maximal length of consecutive $i$’s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_n(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$D\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\frac{1}{log{\theta }_i}\right\}$$ is $1$, where ${\theta }_i=1+\sqrt{4i+1}$. We also obtain that the level set $$E_{\alpha }\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\alpha \right\}$$ is of full Hausdorff dimension...