Subshift Research Articles

Sufficient condition for a topological self-similar set to be a self-similar set

A self-similar set always possesses a self-similar topological structure coded by the shift space (symbolic space), which is considered as the coordinate system for this set. On the contrary, it is known that given a compact set K with self-similar topological structure, there may not exist a metric d such that (K,d) is a self-similar set with the same topological structure. We provide an easy-to-use sufficient condition for the existence of such metric d in terms of the associated graph with respect to the self-similar topological structure. Therefore, one can easily construct a required self-similar set from the shift space by specifying the topological structure.

Minimal and proximal examples of -stable and -approachable shift spaces

Abstract We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ( $\bar {d}$ -approachable shift spaces). The class of $\bar {d}$ -approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$ -approachability, together with a closely connected notion of $\bar {d}$ -shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$ -approachability and $\bar {d}$ -shadowing. Here, we study further properties and connections between $\bar {d}$ -shadowing and $\bar {d}$ -approachability. We prove that $\bar {d}$ -shadowing implies $\bar {d}$ -stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$ -shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$ -shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$ -shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$ -shadowing indeed generalizes the specification property.

Equilibrium measures for two-sided shift spaces via dimension theory

Abstract Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them

Abstract The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of Hölder potentials globally, which answers a question of Kucherenko and Quas. In the second part of the work, we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as the number of symbols in the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.

Decidable problems in substitution shifts

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the shift space generated by iteration, such as aperiodicity, recognizability and (under an additional assumption) irreducibility, or minimality.

Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs

Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs

A Generalizable Method for Practical Non-Intrusive Load Monitoring via Metric-Based Meta-Learning

Non-Intrusive Load Monitoring (NILM) has become a research boom due to its energy-saving benefits as the smart grid develops and the greenhouse effect intensifies. However, existing NILM models may perform poorly in the target domains collected from different sources, or those containing unseen appliance types, blocking the rollout of NILM. In contrast to the methods that up-date the trained model on the unseen data to handle domain shift or heterogeneous label space, this paper proposes a novel general-izable NILM method based on metric-based meta-learning and su-pervised pre-training. It beforehand learns the cross-domain meta-knowledge through meta-training to eliminate the need for parameter updates in the target domain. Moreover, this method is specifically designed for practical deployments at the edge where both labeled instances and computation resources are limited. The proposed method transforms the appliance current into a metric vector via lightweight 1D convolution and then uses the relation module to propagate the labels via similarity. The experimental results on three public datasets and the proposed dataset demon-strate that the proposed method can achieve high generalization performance in different datasets, users houses, appliance brands, and appliance types.

State-space model realization for non-commensurate fractional-order systems based on Gleason’s problem

This paper presents a novel state-space model (SSM) realization approach for non-commensurate fractional-order (NCFO) systems. A new notion of the resolvent invariant backward shift space of Gleason’s problem associated with the NCFO rational transfer function matrix is introduced firstly, based on which a necessary and sufficient condition for the existence of the NCFO-SSM realization is given. Then, a sufficient condition for the construction of the resolvent invariant backward shift space of NCFO Gleason’s problem is presented, and the corresponding realization algorithms are developed. In order to obtain a lower-dimensional NCFO-SSM, further improvement based on a polynomial description is studied. Finally, symbolic and numerical examples are provided to illustrate the details as well as the effectiveness of the proposed NCFO-SSM realization approach.

Entropy of Local Homeomorphisms With Applications to Infinite Alphabet Shift Spaces

Abstract In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu–Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew’s definition of entropy via covers and Bowen’s definition of entropy via separated sets. We propose a definition of factor map between Deaconu–Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.

Asymptotic solution of Bowen equation for perturbed potentials on shift spaces with countable states

In this paper, we study the asymptotic expansions for the zero of the pressure function s\mapsto P(s\varphi(\varepsilon,\cdot)+\xi(\varepsilon,\cdot)) for perturbed potentials \varphi(\varepsilon,\cdot) and \xi(\varepsilon,\cdot) defined on the shift space with countable state space. In our main result, we give a sufficient condition for the solution s=s(\varepsilon) of P(s\varphi(\varepsilon,\cdot)+\xi(\varepsilon,\cdot))=0 to have the n -order asymptotic expansion for the small parameter \varepsilon . In addition, we also obtain the case where the order of the expansion of the solution s=s(\varepsilon) is less than the order of the expansion of the perturbed potentials. Our results can be applied to problems concerning asymptotic behaviours of Hausdorff dimensions given by the Bowen formula: conformal graph directed Markov systems and other concrete examples.

Decidability of CPC-irreducibility of subshifts of finite type over free groups

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the ({textsf{d}},{textsf{c}})-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.

Invariant measures in non-conformal fibered systems with singularities

We study a large class of transformations FT which are only piecewise differentiable on countably many pieces and also non-conformal. There exist random countably generated limit sets JT,ω in the fibers of FT. We prove a Global Volume Lemma implying that the push-forward measures of equilibrium measures from a countable shift space, are exact dimensional on the non-compact global basic set JT. A dimension formula of Ledrappier-Young type is obtained for these measures, by using their Lyapunov exponents and marginal entropies. Then, we study the geometric potentials ψT,s on ΣI, and we prove that the dimensions of the push-forward measures νsω in fibers are independent of ω, and they depend real-analytically on the parameter s from an interval F(T). Moreover, we establish a Variational Principle for dimension in fibers. Our results apply in particular to new types of multidimensional continued fractions.

A note on the nuclear dimension of Cuntz–Pimsner -algebras associated with minimal shift spaces

AbstractFor every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz–Pimsner $C^*$ -algebra $\mathcal {O}_X$ has nuclear dimension $1$ when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X, which also recovers an exact sequence, discovered before by Carlsen and Eilers.

Crafting literary Urdu: Mirza Hatim’s engagement with Vali Dakhani

AbstractThe emergence in eighteenth-century India of literary compositions that used the elite registers of what was, at the time, called ‘Rekhtah’, and later defined as Urdu, is poorly understood. Conventionally, after an initial infatuation in Delhi with the works of Vali Dakhani,1 a mid-century break is assumed, exemplified by the revision of Zuhur ud-Din Hatim’s Divan as Divanzadah in the 1750s. Scholars have viewed this as a radical intervention in the creation of Urdu, which excised old vernacular models and embraced further Persianization. This article re-examines the evidence, combining methodologies from literary and historical studies. It points to the continuities present in Hatim’s revision, including sustained engagement with Vali, even as Hatim attempted to appeal to new audiences, incorporating new trends alongside older literary models. Foregrounding literary networks and arenas of poetic practice shows the limited impact of the proscriptions and literary criticisms voiced by Hatim’s critics. In studying the contested space of literary aesthetics and linguistic shifts against self-fashioning within changing networks, this article demonstrates that the relationship between the Persianate and vernacular sphere continued to be generative, rather than oppositional or hierarchical.

There is no largest proper operator ideal

An operator ideal is proper if the only invertible operators it contains have finite rank. We answer a problem posed by Pietsch (Operator ideals, North-Holland, Amsterdam, 1980) by proving (i) that the ideal of inessential operators is not maximal among proper operator ideals, and (ii) that there is no largest proper operator ideal. Our proof is based on an extension of the construction by Aiena and González (Math Z 233:471–479, 2000), of an improjective but essential operator on Gowers–Maurey’s shift space \(X_S\) (Math Ann 307:543–568, 1997), through a new analysis of the algebra of operators on powers of \(X_S\). We also prove that certain properties hold for general \(\mathbb {C}\)-linear operators if and only if they hold for these operators seen as real: for example this holds for operators belonging to the ideals of strictly singular, strictly cosingular, or inessential operators, answering a question of González and Herrera (Stud Math 183(1):1–14, 2007). This gives us a frame to extend the negative answer to the problem of Pietsch to the real setting.

Hausdorff dimension of frequency sets in beta-expansions

By applying a recent result on the distributions of full and non-full words, we give a proof of the useful folklore result: the Hausdorff dimension of an arbitrary set in the shift space is equal to the Hausdorff dimension of its natural projection in [0, 1]. It has been used in some former papers without proof. Then we clarify that in order to calculate the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit step when the $$\beta $$ -expansion of 1 is finite. Finally, as an application, we obtain an exact formula for the Hausdorff dimension of frequency sets for an important class of $$\beta $$ ’s, which are called pseudo-golden ratios (also called multinacci numbers).

Suberesolving codes

‎We show that any right continuing factor code with retract 0 into an irreducible shift of finite type is right eresolving‎, ‎and we give some sufficient conditions for a right eresolving almost everywhere code being right eresolving everywhere‎. ‎Suberesolving codes as a generalization of ersolving codes have been introduced and we determine some shift spaces which preserved by suberesolving codes‎. ‎Also‎, ‎we show that any bi-eresolving (resp‎. ‎bi-suberesolving) code on an irreducible shift of finite type (resp‎. ‎a synchronized system) is open (resp‎. ‎semi-open) and any right suberesolving code on a synchronized system is right continuing almost everywhere‎.

KMS states and continuous orbit equivalence for ultragraph shift spaces with sinks

We extend ultragraph shift spaces and the realization of ultragraph C*-algebras as partial crossed products to include ultragraphs with sinks (under a mild condition, called (RFUM2), which allow us to dismiss the use of filters) and we describe the associated transformation groupoid. Using these characterizations we study continuous orbit equivalence of ultragraph shift spaces (via groupoids) and KMS and ground states (via partial crossed products).

Computability of topological pressure on compact shift spaces beyond finite type* *Burr was partially supported by grants from the National Science Foundation (CCF-1527193 and DMS-1913119). Das was partially supported by ONR MURI Grant N00014-19-1-242 and ONR YIP Grant N00014-16-1-264. Wolf was partially supported by grants from the Simons Foundation (#637594 to Christian Wolf) and PSC-CUNY (TRADB-51-63715 to Christian Wolf). Yang

We investigate the computability (in the sense of computable analysis) of the topological pressure P top(ϕ) on compact shift spaces X for continuous potentials . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X, ϕ) ↦ P top(X, ϕ| X ) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X ↦ h top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

Topologically Mixing Properties of Multiplicative Integer Systems

Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $\mathsf{X}_\Omega^{(\ell)}$ is the multiplicative subshift derived from the shift space $\Omega$ with given $\ell > 1$. We show that $\mathsf{X}_\Omega^{(\ell)}$ is (topologically) transitive/mixing if and only if $\Omega$ is extensible/mixing. After introducing $\ell$-directional mixing property, we derive the equivalence between $\ell$-directional mixing property of $\mathsf{X}_\Omega^{(\ell)}$ and weakly mixing property of $\Omega$.