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Tuomas Sahlsten

Publications and source records attributed to Tuomas Sahlsten.

31 records · Page 2Linked to original sources
arXiv2017

On the Fourier analytic structure of the Brownian graph

In a previous article (\textit{Int. Math. Res. Not.} 2014, 2730--2745) T. Orponen and the authors proved that the Fourier dimension of the graph of any real-valued function on $\mathbb{R}$ is bounded above by $1$. This partially answered a question of Kahane ('93) by showing that the graph of the Wiener process $W_t$ (Brownian motion) is almost surely not a Salem set. In this article we complement this result by showing that the Fourier dimension of the graph of $W_t$ is almost surely $1$. In the proof we introduce a method based on Ito calculus to estimate Fourier transforms by reformulating the question in the language of Ito drift-diffusion processes and combine it with the classical work of Kahane on Brownian images.

math.PR↗
arXiv2017

Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

math.SP↗
arXiv2016

Pointwise perturbations of countable Markov maps

We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let $T_k$ and $T$ be expanding countable Markov maps such that the inverse branches of $T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then under suitable regularity assumptions on the maps $T_k$ and $T$ the following limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : θ_k'(x) \neq 0\} = 1,$$ where $θ_k$ is the topological conjugacy between $T_k$ and $T$ and $\dim_\mathrm{H}$ stands for the Hausdorff dimension. This is in contrast with the fact that other natural quantities measuring the singularity of $θ_k$ fail to be continuous in this manner under pointwise convergence such as the Hölder exponent of $θ_k$ or the Hausdorff dimension $\dim_\mathrm{H} (μ\circ θ_k)$ for the preimage of the absolutely continuous invariant measure $μ$ for $T$. As an application we obtain a perturbation theorem in non-uniformly hyperbolic dynamics for conjugacies between intermittent Manneville-Pomeau maps $x \mapsto x + x^{1+α} \mod 1$ when varying the parameter $α$.

math.DS↗
arXiv2015

Intermediate β-shifts of finite type

An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $β$-shift (transformation) and an intermediate $β$-shift (transformation), for a fixed $β\in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{β,α} \colon x \mapsto βx + α\bmod 1$ for which the corresponding intermediate $β$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(β,α)$ such that the intermediate $β$-shift associated with $T_{β, α}$ is a subshift of finite type. It is also proved that these maps $T_{β,α}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $β$-shifts and $β$-transformations, for which both of the two properties do not hold.

math.DS↗
arXiv2015

Structure of distributions generated by the scenery flow

We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery flow. We further show that the collection of fractal distributions is closed under the weak topology and, moreover, is a Poulsen simplex, that is, extremal points are dense. We apply these to show that a Baire generic measure is as far as possible from being uniformly scaling: at almost all points, it has all fractal distributions as tangent distributions.

math.DS↗
arXiv2015

Dynamics of the scenery flow and geometry of measures

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

math.CA↗
arXiv2015

Fourier transforms of Gibbs measures for the Gauss map

We investigate under which conditions a given invariant measure $μ$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehatμ(ξ)| = O(|ξ|^{-η}), \quad \text{as } |ξ| \to \infty.$$ We show that this property holds for any Gibbs measure $μ$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queffélec-Ramaré. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erdős-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.

math.DS↗
arXiv2014

Tangent measures of typical measures

We prove that for a typical Radon measure mu in R^d, every non-zero Radon measure is a tangent measure of mu at mu almost every point. This was already shown by T. O'Neil in his PhD thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.

math.CA↗
arXiv2014

Scaling scenery of $(\times m,\times n)$ invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism $(x,y) \mapsto (mx \mod 1,ny \mod 1)$ that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman-Shmerkin and Ferguson-Jordan-Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford-McMullen, Lalley-Gatzouras and Barański class and all planar self-similar sets.

math.DS↗
arXiv2013

On Fourier analytic properties of graphs

We study the Fourier dimensions of graphs of real-valued functions defined on the unit interval [0,1]. Our results imply that the graph of the fractional Brownian motion is almost surely not a Salem set, answering in part a question of Kahane from 1993, and that the graph of a Baire typical function in C[0,1] has Fourier dimension zero.

math.CA↗
arXiv2012

Dimension, entropy, and the local distribution of measures

We present a general approach to the study of the local distribution of measures on Euclidean spaces, based on local entropy averages. As concrete applications, we unify, generalize, and simplify a number of recent results on local homogeneity, porosity and conical densities of measures.

math.CA↗
arXiv2011

Radial projections of rectifiable sets

We show that if no $m$-plane contains almost all of an $m$-rectifiable set $E \subset \R^{n}$, then there exists a single $(m - 1)$-plane $V$ such that the radial projection of $E$ has positive $m$-dimensional measure from every point outside $V$.

math.CA↗