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Yonatan Gutman

Publications and source records attributed to Yonatan Gutman.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2020

Mean dimension and an embedding theorem for real flows

We develop mean dimension theory for $\mathbb{R}$-flows. We obtain fundamental properties and examples and prove an embedding theorem: Any real flow $(X,\mathbb{R})$ of mean dimension strictly less than $r$ admits an extension $(Y,\mathbb{R})$ whose mean dimension is equal to that of $(X,\mathbb{R})$ and such that $(Y,\mathbb{R})$ can be embedded in the $\mathbb{R}$-shift on the compact function space $\{f\in C(\mathbb{R},[-1,1])|\;\mathrm{supp}(\hat{f})\subset [-r,r]\}$, where $\hat{f}$ is the Fourier transform of $f$ considered as a tempered distribution. These canonical embedding spaces appeared previously as a tool in embedding results for $\mathbb{Z}$-actions.

math.DS↗
arXiv2020

The structure theory of Nilspaces III: Inverse limit representations and topological dynamics

This paper forms the third part of a series by the authors [GMV1,GMV2] concerning the structure theory of nilspaces. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$, satisfying some natural axioms. Our goal is to extend the structure theory of nilspaces obtained by Antolín Camarena and Szegedy, and to provide new proofs. Our main result is that, under the technical assumption that $C^n(X)$ is a connected space for all $n$, then $X$ is isomorphic (in a strong sense) to an inverse limit of nilmanifolds. This is a direct and slight generalization of the main result of Antolín Camarena and Szegedy. We also apply our methods to obtain structure theorems in the setting of topological dynamics. Specifically, if $H$ is a group (subject to very mild topological assumptions) and $(H,X)$ is a minimal dynamical system, then we give a simple characterization of the maximal pronilfactor of $X$. This generalizes the case $H = \mathbb{Z}$, which is a theorem of Host, Kra and Maass, although even in that case we give a significantly different proof.

math.DS↗
arXiv2019

New Uniform Bounds for Almost Lossless 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 $\mathcal{S} \subset [0,1]^\mathbb{Z}$ 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↗
arXiv2018

An explicit compact universal space for real flows

The Kakutani-Bebutov Theorem (1968) states that any compact metric real flow whose fixed point set is homeomorphic to a subset of $\mathbb{R}$ embeds into the Bebutov flow, the $\mathbb{R}$-shift on $C(\mathbb{R},[0,1])$. An interesting fact is that this universal space is a function space. However, it is not compact, nor locally compact. We construct an explicit countable product of compact subspaces of the Bebutov flow which is a universal space for all compact metric real flows, with no restriction; namely, into which any compact metric real flow embeds. The result is compared to previously known universal spaces.

math.DS↗
arXiv2018

The structure theory of Nilspaces I

This paper forms the first part of a series by the authors [GMV2,GMV3] concerning the structure theory of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes $C^n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$ satisfying some natural axioms. Antolín Camarena and Szegedy proved that from these axioms it follows that (certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of nilmanifolds. The aim of our project is to provide a new self-contained treatment of this theory and give new applications to topological dynamics. This paper provides an introduction to the project from the point of view of applications to higher order Fourier analysis. We define and explain the basic definitions and constructions related to cubespaces and nilspaces and develop the weak structure theory, which is the first stage of the proof of the main structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. We also make some modest innovations and extensions to this theory. In particular, we consider a class of maps that we term fibrations, which are essentially equivalent to what are termed fiber-surjective morphisms by Anatolín Camarena and Szegedy, and we formulate and prove a relative analogue of the weak structure theory alluded to above for these maps. These results find applications elsewhere in the project.

math.DS↗
arXiv2018

Higher order regionally proximal equivalence relations for general minimal group actions

We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these relations are equivalence relations whenever the action is minimal. This was known for abelian actions by a result of Shao and Ye. We also show that these relations lift through extensions between minimal systems. Answering a question by Tao, given a minimal system, we prove that the regionally proximal equivalence relation of order $d$ corresponds to the maximal dynamical Antolín Camarena--Szegedy nilspace factor of order at most $d$. In particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor. Finally by using a result of Gutman, Manners and Varjú under some restrictions on the acting group, it follows that the regionally proximal equivalence relation of order $d$ corresponds to the maximal pronilfactor of order at most $d$ (a factor which is an inverse limit of nilsystems of order at most $d$).

math.DS↗
arXiv2017

The structure theory of Nilspaces II: Representation as nilmanifolds

This paper forms the second part of a series by the authors [GMV1,GMV3] concerning the structure theory of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space $X$ together with closed collections of cubes $C_n(X)\subseteq X^{2^n}$, $n=1,2,\ldots$ satisfying some natural axioms. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antolín Camarena and Szegedy [CS12], stating that if each of these groups is a torus then $X$ is isomorphic (in a strong sense) to a nilmanifold $G/Γ$. We also extend the theorem to a setting where the nilspace arises from a dynamical system $(X,T)$. These theorems are a key stepping stone towards the general structure theorem in [GMV3] (which again closely resembles the main theorem of [CS12]). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in [CS12].

math.DS↗
arXiv2017

The embedding problem in topological dynamics and Takens' theorem

We prove that every $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,\mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,σ\times S)$, where $D\in\mathbb{N}$, $L\in\mathbb{N}\cup\{0\}$ and $σ$ is the shift on the Hilbert cube $([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$; in particular, when $(Y,\mathbb{Z}^{k},S)$ is an irrational $\mathbb{Z}^{k}$-rotation on the $k$-torus, $(X,\mathbb{Z}^{k},T)$ embeds in $(([0,1]^{2^kD+1})^{\mathbb{Z}^k},σ)$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens' embedding theorem with a continuous observable for $\mathbb{Z}$-actions and deduce the analogous result for $\mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous conjecture for $\mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for $\mathbb{Z}^{k}$-actions on finite dimensional spaces.

math.DS↗
arXiv2017

Application of signal analysis to the embedding problem of $\mathbb{Z}^k$-actions

We study the problem of embedding arbitrary $\mathbb{Z}^k$-actions into the shift action on the infinite dimensional cube $\left([0,1]^D\right)^{\mathbb{Z}^k}$. We prove that if a $\mathbb{Z}^k$-action satisfies the marker property (in particular if it is a minimal system without periodic points) and if its mean dimension is smaller than $D/2$ then we can embed it in the shift on $\left([0,1]^D\right)^{\mathbb{Z}^k}$. The value $D/2$ here is optimal. The proof goes through signal analysis. We develop the theory of encoding $\mathbb{Z}^k$-actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of the Euclidean space into a band-limited function which is constructed from another tiling.

math.DS↗
arXiv2017

Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.

math.DS↗
arXiv2016

Takens' embedding theorem with a continuous observable

Let $(X,T)$ be a dynamical system where $X$ is a compact metric space and $T:X\rightarrow X$ is continuous and invertible. Assume the Lebesgue covering dimension of $X$ is $d$. We show that for a generic continuous map $h:X\rightarrow[0,1]$, the $(2d+1)$-delay observation map $x\mapsto\big(h(x),h(Tx),\ldots,h(T^{2d}x)\big)$ is an embedding of $X$ inside $[0,1]^{2d+1}$. This is a generalization of the discrete version of the celebrated Takens embedding theorem, as proven by Sauer, Yorke and Casdagli to the setting of a continuous observable. In particular there is no assumption on the (lower) box-counting dimension of $X$ which may be infinite.

math.DS↗
arXiv2015

Embedding minimal dynamical systems into Hilbert cubes

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $\left([0,1]^N\right)^{\mathbb{Z}}$. This problem is intimately related to the theory of mean dimension, which counts the averaged number of parameters of dynamical systems. Lindenstrauss proved that minimal systems of mean dimension less than $N/36$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$, and he proposed the problem of finding the optimal value of the mean dimension for the embedding. We solve this problem by proving that minimal systems of mean dimension less than $N/2$ can be embedded into $\left([0,1]^N\right)^{\mathbb{Z}}$. The value $N/2$ is optimal. The proof uses Fourier and complex analysis.

math.DS↗
arXiv2015

Mean Dimension & Jaworski-type Theorems

According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim(X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1]^{d+1})^{\mathbb{Z}},shift)$.

math.DS↗
arXiv2015

Mean dimension of $\mathbb{Z}^k$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $\mathbb{Z}^k$-action on a compact metric space $X$, we study the following three problems closely related to mean dimension. (1) When is $X$ isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy $h_{\mathrm{top}}(X)$ is infinite. How much topological entropy can be detected if one considers $X$ only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed $X$ into the $\mathbb{Z}^k$-shift on the infinite dimensional cube $([0,1]^D)^{\mathbb{Z}^k}$? These were investigated for $\mathbb{Z}$-actions in [Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math. \textbf{89} (1999) 227-262], but the generalization to $\mathbb{Z}^k$ remained an open problem. When $X$ has the marker property, in particular when $X$ has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3). A key ingredient is a new method to continuously partition every orbit into good pieces.

math.DS↗
arXiv2015

On relative extreme amenability

The purpose of this paper is to study the notion of relative extreme amenability for pairs of topological groups. We give a characterization by a fixed point property on universal spaces. In addition we introduce the concepts of an extremely amenable interpolant as well as maximally relatively extremely amenable pairs and give examples. It is shown that relative extreme amenability does not imply the existence of an extremely amenable interpolant. The theory is applied to generalize results of Kechris, Pestov and Todorcevic relating to the application of Fraïssé theory to the theory of Dynamical Systems. In particular, new conditions enabling to characterize universal minimal spaces of automorphism groups of Fraïssé structures are given.

math.GR↗