arXiv subjects

Adam Śpiewak

Publications and source records attributed to Adam Śpiewak.

17 recordsLinked to original sources

arXiv2026

Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals

We introduce the mean Assouad dimension of a dynamical system, motivated by the Assouad dimension in fractal geometry. Using dimension interpolation, we further define the mean Assouad spectrum. This provides a new family of bi-Lipschitz invariants of dynamical systems. We study its basic properties and calculate it for several classes of dynamical systems. As an application, we determine explicit formulae for the mean Assouad dimension and spectrum of infinite-dimensional Bedford--McMullen carpet systems, contributing to the program of studying infinite dimensional fractals, initiated recently by Tsukamoto.

math.DS↗

arXiv2025

Optimal compressed sensing for mixing stochastic processes

Jalali and Poor introduced an asymptotic framework for compressed sensing of stochastic processes, demonstrating that any rate strictly greater than the mean information dimension serves as an upper bound on the number of random linear measurements required for (universal) almost lossless recovery of $\psi^*$-mixing processes, as measured in the normalized $L^2$ norm. In this work, we show that if the normalized number of random linear measurements is strictly less than the mean information dimension, then almost lossless recovery of a $\psi^*$-mixing process is impossible by any sequence of decompressors. This establishes the mean information dimension as the fundamental limit for compressed sensing in this setting (and, in fact, the precise threshold for the problem). To this end, we introduce a new quantity, related to techniques from geometric measure theory: the correlation dimension rate, which is shown to be a lower bound for compressed sensing of arbitrary stationary stochastic processes.

cs.IT↗

arXiv2025

On the regularity of time-delayed embeddings with self-intersections

We study regularity of the time-delayed coordinate maps \[\phi_{h,k}(x) = (h(x), h(Tx), \ldots, h(T^{k-1}x))\] for a diffeomorphism $T$ of a compact manifold $M$ and smooth observables $h$ on $M$. Takens' embedding theorem shows that if $k > 2\dim M$, then $\phi_{h,k}$ is an embedding for typical $h$. We consider the probabilistic case, where for a given probability measure $\mu$ on $M$ one allows self-intersections in the time-delayed embedding to occur along a zero-measure set. We show that if $k \geq \dim M$ and $k > \dim_H(\text{supp} \mu)$, then for a typical observable, $\phi_{h,k}$ is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover $k > \dim M$, then $\phi_{h,k}$ is a local diffeomorphism at almost every point. As an application, we show that if $k > \dim M$, then the Lyapunov exponents of the original system can be approximated with arbitrary precision by almost every orbit in the time-delayed model of the system. We also give almost sure pointwise bounds on the prediction error and provide a non-dynamical analogue of the main result, which can be seen as a probabilistic version of Whitney's embedding theorem.

math.DS↗

arXiv2024

Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously

We prove a universal projection theorem, giving conditions on a parametrized family of maps $\Pi_\lambda : X \to \mathbb{R}^d$ and a collection M of measures on X under which for almost every $\lambda$ equality $\dim_H \Pi_\lambda \mu = \min\{d, \dim_H \mu\}$ holds for all measures $\mu \in M$ simultaneously (i.e. on a full measure set of $\lambda$'s independent of $\mu$). We require family $\Pi_\lambda$ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every $\lambda$, projection $\Pi_\lambda$ is nearly bi-Lipschitz (i.e. with pointwise $\alpha$-H\"older inverse for every $\alpha \in (0,1)$) at $\mu$-a.e. x, for all $\mu \in M$ simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS. As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part). Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation).

math.DS↗

arXiv2024

Typical dimension and absolute continuity for classes of dynamically defined measures, Part II : exposition and extensions

This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of ergodic measures for parametrized families of smooth iterated function systems (IFS) on the line. Under some assumptions, most crucially, a transversality condition, we obtain formulas for the Hausdorff dimension of the measure and absolute continuity for almost every parameter in the appropriate parameter region. The main novelty of [1] and the present paper is that not only the IFS, but also the ergodic measure in the symbolic space, whose push-forward we consider, depends on the parameter. This includes many interesting families of measures, in particular, invariant measures for IFS's with place-dependent probabilities and natural (equilibrium) measures for smooth IFS's. One of the goals of this paper is to present an exposition of [1] in a more reader-friendly way, emphasizing the ideas and proof strategies, but omitting the more technical parts. This exposition/survey is based in part on the series of lectures by K\'aroly Simon at the Summer School "Dynamics and Fractals" in 2023 at the Banach Center, Warsaw. The main new feature, compared to [1], is that we consider multi-parameter families; in other words, the set of parameters is allowed to be multi-dimensional. This broadens the scope of applications. A new application considered here is to a class of Furstenberg-like measures. [1] B. B\'ar\'any, K. Simon, B. Solomyak and A. \'Spiewak: Typical absolute continuity for classes of dynamically defined measures. Advances in Mathematics, Volume 399, 2022, 108258, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108258.

math.DS↗

arXiv2024

Predicting dynamical systems with too few time-delay measurements: error estimates

We study the problem of reconstructing and predicting the future of a dynamical system by the use of time-delay measurements of typical observables. Considering the case of too few measurements, we prove that for Lipschitz systems on compact sets in Euclidean spaces, equipped with an invariant Borel probability measure $\mu$ of Hausdorff dimension $d$, one needs at least $d$ measurements of a typical (prevalent) Lipschitz observable for $\mu$-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of $\mu$ is the precise threshold for the minimal delay (embedding) dimension for such systems in a probabilistic setting. Furthermore, we establish a lower bound postulated in the Schroer--Sauer--Ott--Yorke prediction error conjecture from 1998, after necessary modifications (whereas the upper estimates were obtained in our previous work). To this aim, we prove a general theorem on the dimensions of conditional measures of $\mu$ with respect to time-delay coordinate maps.

math.DS↗

arXiv2023

Regularity of almost-surely injective projections in Euclidean spaces

In a previous work we proved that if a finite Borel measure $\mu$ in a Euclidean space has Hausdorff dimension smaller than a positive integer $k$, then the orthogonal projection onto almost every $k$-dimensional linear subspace is injective on a set of full $\mu$-measure. In this paper we study the regularity of the inverses of these projections and prove that if $\mu$ has a compact support $X$ such that (respectively) the Hausdorff, upper box-counting or Assouad dimension of $X$ is smaller than $k$, then the inverse is (respectively) continuous, pointwise $\alpha$-H\"older for some $\alpha \in (0,1)$ or pointwise $\alpha$-H\"older for every $\alpha \in (0,1)$. The results generalize to the case of typical linear perturbations of Lipschitz maps and strengthen previously known ones in the lossless analog compression literature. We provide examples showing the sharpness of the statements. Additionally, we construct a non-trivial measure on the plane which admits almost-surely injective projections in every direction, and show that no homogeneous self-similar measure has this property.

math.CA↗

arXiv2023

Absolute continuity of self-similar measures on the plane

Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in \mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every choice of $(\lambda_1, \ldots, \lambda_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.

math.DS↗

arXiv2022

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism $T$ of a Riemannian manifold $X$ admits an attractor with a natural measure $\mu$ of information dimension smaller than $k$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for $\mu$-almost sure prediction of future measurements of $h$. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations $T$ of a compact set $X$ in Euclidean space with an ergodic $T$-invariant Borel probability measure $\mu$. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or H\"older systems on Borel sets in Euclidean space.

math.DS↗

arXiv2022

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsed\`a--Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM-system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsed\`a and Misiurewicz from 2014.

math.DS↗

arXiv2021

On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings

Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if $\mu$ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and $k$ is greater than the information dimension of $\mu$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for a predictable reconstruction of $\mu$-almost every initial point of the original system. This reduces by half the number of required measurements, compared to the standard (deterministic) setup. We prove the conjecture for ergodic measures and show that it holds for a generic smooth diffeomorphism, if the information dimension is replaced by the Hausdorff one. To this aim, we prove a general version of predictable embedding theorem for injective Lipschitz maps on compact sets and arbitrary Borel probability measures. We also construct an example of a $C^\infty$-smooth diffeomorphism with a natural measure, for which the conjecture does not hold in its original formulation.

math.DS↗

arXiv2021

Typical absolute continuity for classes of dynamically defined measures

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^\lambda_j\}$ on the real line. Given a family of parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we study geometric and dimensional properties of their images under the natural projection maps $\Pi^\lambda$. The main novelty of our work is that the measures $\mu_\lambda$ depend on the parameter, whereas up till now it has been usually assumed that the measure on the symbolic space is fixed and the parameter dependence comes only from the natural projection. This is especially the case in the question of absolute continuity of the projected measure $(\Pi^\lambda)_*\mu_\lambda$, where we had to develop a new approach in place of earlier attempt which contains an error. Our main result states that if $\mu_\lambda$ are Gibbs measures for a family of H\"older continuous potentials $\phi^\lambda$, with H\"older continuous dependence on $\lambda$ and $\{\Pi^\lambda\}$ satisfy the transversality condition, then the projected measure $(\Pi^\lambda)_*\mu_\lambda$ is absolutely continuous for Lebesgue a.e.\ $\lambda$, such that the ratio of entropy over the Lyapunov exponent is strictly greater than $1$. We deduce it from a more general almost sure lower bound on the Sobolev dimension for families of measures with regular enough dependence on the parameter. Under less restrictive assumptions, we also obtain an almost sure formula for the Hausdorff dimension. As applications of our results, we study stationary measures for iterated function systems with place-dependent probabilities (place-dependent Bernoulli convolutions and the Blackwell measure for binary channel) and equilibrium measures for hyperbolic IFS with overlaps (in particular: natural measures for non-homogeneous self-similar IFS and certain systems corresponding to random continued fractions).

math.DS↗

arXiv2020

Around the variational principle for metric mean dimension

We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second we derive a variational principle for metric mean dimension involving growth rates of measure-theoretic entropy of partitions decreasing in diameter which holds in full generality and in particular does not necessitate the assumption of tame growth of covering numbers. The expressions involved are a dynamical version of Renyi information dimension. Third we derive a new expression for Geiger-Koch information dimension rate for ergodic shift-invariant measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.

math.DS↗

arXiv2019

New Uniform Bounds for Almost Lossless Analog Compression

Wu and Verd\'u developed a theory of almost lossless analog compression, where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a (typically infinite-entropy) stationary stochastic process. In this work we consider all stationary stochastic processes with trajectories in a prescribed set $\mathcal{S} \subset [0,1]^\mathbb{Z}$ of (bi)infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension. An essential tool is the recent Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension in terms of rate-distortion functions.

math.DS↗

arXiv2019

Singular stationary measures for random piecewise affine interval homeomorphisms

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.

math.DS↗

arXiv2018

Metric mean dimension and analog compression

Wu and Verd\'u developed a theory of almost lossless analog compression, where one imposes various regularity conditions on the compressor and the decompressor with the input signal being modelled by a (typically infinite-entropy) stationary stochastic process. In this work we consider all stationary stochastic processes with trajectories in a prescribed set of (bi-)infinite sequences and find uniform lower and upper bounds for certain compression rates in terms of metric mean dimension and mean box dimension. An essential tool is the recent Lindenstrauss-Tsukamoto variational principle expressing metric mean dimension in terms of rate-distortion functions. We obtain also lower bounds on compression rates for a fixed stationary process in terms of the rate-distortion dimension rates and study several examples.

math.DS↗

arXiv2018

A probabilistic Takens theorem

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 $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a given Lipschitz map $h : X \to \mathbb{R}$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $\mu$. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, B\"{o}lcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required measurements from $2\dim X$ to $\dim X$ and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

math.DS↗