arXiv subjects
Publications and source records attributed to Stéphane Seuret.
Any ergodic measure of a smooth map on a compact manifold has a multifractal spectrum with one point - the dimension of the measure itself - at the diagonal. We will construct examples where this fails in the most drastic way for invariant measures invariant under linear maps of the circle.
For any $s\in (1/2,1]$, the series$F_s(x)=\sum_{n=1}^{\infty} e^{iπn^2 x}/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in terms of even or regular continued fractions certain subsets of points of $[-1,1]$ of full measure where the series converges. Our method is based on an approximate function equation for $F_s(x)$. As a by-product, we obtain the convergence of certain series defined in term of the convergents of the even continued fraction of an irrational number.
We introduce a local multifractal formalism adapted to functions, measures or distributions which display multifractal characteristics that can change with time, or location. We develop this formalism in a general framework and we work out several examples of measures and functions where this setting is relevant.
We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space $Σ_m=\{0,...,m-1\}^\N$ that are invariant under multiplication by integers. The results apply to the sets $\{x\in Σ_m: \forall\, k, \ x_k x_{2k}... x_{n k}=0\}$, where $n\ge 3$. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.
In this article, we prove that in the Baire category sense, measures supported by the unit cube of $\R^d$ typically satisfy a multifractal formalism. To achieve this, we compute explicitly the multifractal spectrum of such typical measures $μ$. This spectrum appears to be linear with slope 1, starting from 0 at exponent 0, ending at dimension $d$ at exponent $d$, and it indeed coincides with the Legendre transform of the $L^q$-spectrum associated with typical measures $μ$.
Let $1 \leq d < D$ and $(p,q,s)$ satisfying $0 < p < \infty$, $0 < q \leq \infty$, $0 < s-d/p < \infty$. In this article we study the global and local regularity properties of traces, on affine subsets of $\R^D$, of functions belonging to the Besov space $B^{s}_{p,q}(\R^D)$. Given a $d$-dimensional subspace $H \subset \R^D$, for almost all functions in $B^{s}_{p,q}(\R^D)$ (in the sense of prevalence), we are able to compute the singularity spectrum of the traces $f_a$ of $f$ on affine subspaces of the form $a+H$, for Lebesgue-almost every $a \in \R^{D-d}$. In particular, we prove that for Lebesgue-almost every $a \in \R^{D-d}$, these traces $f_a$ are more regular than what could be expected from standard trace theorems, and that $f_a$ enjoys a multifractal behavior.