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Yonatan Gutman

Publications and source records attributed to Yonatan Gutman.

At least 19 recordsLinked to original sources
arXiv2026

Dynamical Cantor Staircase Functions and The Small Flow Boundary Property

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, \textit{flow-generated entropy}, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.

math.DS↗
arXiv2026

On the representation of measurable and continuous dynamical systems by Lipschitz functions

Two representations theorems are presented: 1. Any Borel action of a second countable locally compact group $G$ on a standard Borel space $X$ admits an injective $G$-equivariant Borel map into the shift space of $1$-Lipschitz functions from $G$ to the unit interval $Lip_1(G)$. 2. Any continuous action of $\mathbb{R}^k$ ($k\in \mathbb{N}$) on a metrizable compact space $X$ admits an injective $G$-equivariant continuous map into $Lip_1(\mathbb{R}^k)$ if the fixed point set $Fix(X,\mathbb{R}^k)$ embeds into $[0,1]$ and $(X,\mathbb{R}^k)$ is \textit{weakly locally free}, that is $\mathbb{R}^k$ acts freely outside the fixed point set. The first theorem generalizes a theorem from 1973 by Eberlein for $\mathbb{R}$-flows. The second theorem generalizes a Lipschitz refinement of the Bebutov-Kakutani theorem proven by Gutman, Jin and Tsukamoto in 2019.

math.DS↗
arXiv2025

A Lipschitz Refinement of the Multidimensional Bebutov--Kakutani Dynamical Embedding Theorem

We prove that a continuous action of $\mathbb{R}^n$ on a compact metrizable space equivariantly embeds into the shift action on the space of one-Lipschitz functions from $\mathbb{R}^n$ to $[0,1]$ if and only if the set of fixed points topologically embeds in $[0,1]$. This is a Lipschitz refinement of classical dynamical embedding theorems of Bebutov, Kakutani, Jaworski and Chen.

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↗
arXiv2024

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 $μ$ of information dimension smaller than $k$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for $μ$-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 $μ$. 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 Hölder systems on Borel sets in Euclidean space.

math.DS↗
arXiv2024

Local mean dimension theory for sofic group actions

Using a local perspective, we introduce \textit{mean dimension pairs} and give sufficient conditions of when every non-trivial factor of a continuous group action of a sofic group $G$ has positive mean dimension. In addition we show that the mean dimension map is Borel, and that the set of subshifts with completely positive mean dimension of $[0,1]^G$, the full $G$-shift on the interval, is a complete coanalytic set in the set of all subshifts (hence not Borel). Our results are new even when the acting group is $\Z$.

math.DS↗
arXiv2024

Equivariant embedding of finite-dimensional dynamical systems

We prove an equivariant version of the classical Menger-Nobeling theorem regarding topological embeddings: Whenever a group $G$ acts on a finite-dimensional compact metric space $X$, a generic continuous equivariant function from $X$ into $([0,1]^r)^G$ is a topological embedding, provided that for every positive integer $N$ the space of points in $X$ with orbit size at most $N$ has topological dimension strictly less than $\frac{rN}{2}$. We emphasize that the result imposes no restrictions whatsoever on the acting group $G$ (beyond the existence of an action on a finite-dimensional space). Moreover, if $G$ is finitely generated then there exists a finite subset $F\subset G$ so that for a generic continuous map $h:X\to [0,1]^{r}$, the map $h^{F}:X\to ([0,1]^{r})^{F}$ given by $x\mapsto (f(gx))_{g\in F}$ is an embedding. This constitutes a generalization of the Takens delay embedding theorem into the topological category.

math.DS↗
arXiv2024

Strongly isomorphic symbolic extensions for expansive topological flows

In this paper, we prove that finite-dimensional topological flows without fixed points and having a countable number of periodic orbits, have the small flow boundary property. This enables us to answer positively a question of Bowen and Walters from 1972: Any expansive topological flow has a strongly isomorphic symbolic flow extension, i.e. an extension by a suspension flow over a subshift. Previously Burguet had shown this is true if the flow is assumed to be $C^2$-smooth.

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 $μ$ of Hausdorff dimension $d$, one needs at least $d$ measurements of a typical (prevalent) Lipschitz observable for $μ$-almost sure reconstruction and prediction. Consequently, the Hausdorff dimension of $μ$ 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 $μ$ with respect to time-delay coordinate maps.

math.DS↗
arXiv2023

Strictly ergodic distal models and a new approach to the Host-Kra factors

Cocycles are a key object in Antolín Camarena and Szegedy's (topological) theory of nilspaces. We introduce measurable counterparts, named nilcycles, enabling us to give conditions which guarantee that an ergodic group extension of a strictly ergodic distal system admits a strictly ergodic distal topological model, revisiting a problem studied by Lindenstrauss. In particular we show that if the base space is a dynamical nilspace then a dynamical nilspace topological model may be chosen for the extension. This approach combined with a structure theorem of Gutman, Manners and Varjú applied to the ergodic group extensions between successive Host-Kra characteristic factors gives a new proof that these factors are inverse limit of nilsystems.

math.DS↗
arXiv2023

Regularity of almost-surely injective projections in Euclidean spaces

In a previous work we proved that if a finite Borel measure $μ$ 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 $μ$-measure. In this paper we study the regularity of the inverses of these projections and prove that if $μ$ 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 $α$-Hölder for some $α\in (0,1)$ or pointwise $α$-Hölder for every $α\in (0,1)$. 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↗
arXiv2022

Maximal pronilfactors and a topological Wiener-Wintner theorem

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil($k$) systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof we establish that pronilsystems are $coalescent$ both in the measurable and topological categories. In addition, we characterize a CF-Nil($k$) system in terms of its $(k+1)$-$th\ dynamical\ cubespace$. In particular, for $k=1$, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version.

math.DS↗
arXiv2022

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 $μ$ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and $k$ is greater than the information dimension of $μ$, then $k$ time-delayed measurements of a one-dimensional observable $h$ are generically sufficient for a predictable reconstruction of $μ$-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

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↗
arXiv2021

On the structure theory of cubespace fibrations

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over $Y$. If the structure groups of $f$ are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of $f$ are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg's and the Bronstein-Ellis structure theorems in this setting.

math.DS↗
arXiv2020

A probabilistic Takens theorem

Let $X \subset \mathbb{R}^N$ be a Borel set, $μ$ 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 $μ$. 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, Bölcskei, 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↗
arXiv2020

Metric mean dimension and analog compression

Wu and Verdú 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↗