Two-sided Shift Research Articles

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.

Geometrical construction of a chaotic horseshoe-like map on Cantor dust [formula omitted]

Smale’s horseshoe map is one of the most important examples of 2D chaos in dynamical systems. Its chaotic behavior is often illustrated using a two-sided shift map. In this paper, a 3−dimensional horseshoe-like map is defined inspired by the geometrical motivation of the horseshoe map of Smale. Moreover, this construction is expressed on the Cantor dust (C×C×C) via ternary numbers and then the density of the periodic points is proved by using a different technique. To compute these periodic points more rapidly, we also give an algorithm. Finally, it is shown that this version of 3D horseshoe-like map is Devaney chaotic.

Binary factors of shifts of finite type

AbstractWe construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

Dimension of invariant measures for affine iterated function systems

Let {Si}i∈Λ be a finite contracting affine iterated function system (IFS) on Rd. Let (Σ,σ) denote the two-sided full shift over the alphabet Λ, and let π:Σ→Rd be the coding map associated with the IFS. We prove that the projection of an ergodic σ-invariant measure on Σ under π is always exact dimensional, and its Hausdorff dimension satisfies a Ledrappier–Young-type formula. Furthermore, the result extends to average contracting affine IFSs. This completes several previous results and answers a folklore open question in the community of fractals. Some applications are given to the dimension of self-affine sets and measures.

Jaworski-type embedding theorems of one-sided dynamical systems

Embeddings of two-sided dynamical systems in the two-sided shift $(\mathbb R ^{\mathbb Z },\sigma )$ of the real line $\mathbb R $ have been studied by many authors. In this paper, we introduce the notion of trajectory-embedding for the one-sided shift $(

A class of distribution-free one-sided Cucconi schemes for joint surveillance of location and scale parameters and their application in monitoring cab services

Simultaneous distribution-free surveillance (SDFS) of location and scale parameters of a process based on a single plotting statistic has emerged as a prevalent research issue in the past seven years. Most of the SDFS schemes are, however, appropriate for monitoring two-sided shifts in location-scale. In monitoring service quality or time to an event, among others, decreasing process median or variability are usually considered as a process improvement and not something to worry about in practice. We only desire to have a quick signal when there is an increase in the median or scatter as they refer to a deterioration in the process quality. On the contrary, in network traffic surveillance, a decrease in process variability could be a matter of concern as it may result in network congestion where a decreasing shift may be of more interest. In such instances, specific one-sided shifts are of more importance, and a one-sided SDFS scheme would be more appropriate than traditional two-sided schemes. In this article, we propose several Shewhart-Cucconi type SDFS schemes for jointly monitoring one-sided deviations in location or scale or both parameters of a process. We discuss the implementation steps and establish the in-control robustness of the new surveillance plans. We compare proposed schemes with existing one-sided Shewhart-Lepage type schemes using run-length properties. Numerical results substantiate that the proposed schemes perform quite well in many cases. We explain the application of newly designed schemes in surveillance of waiting time during a taxi ride in some cities.

Alternative method to generate three-dimensional (3D) ultrasonographic images in veterinary medicine

Simultaneous distribution-free surveillance (SDFS) of location and scale parameters of a process based on a single plotting statistic has emerged as a prevalent research issue in the past seven years. Most of the SDFS schemes are, however, appropriate for monitoring two-sided shifts in location-scale. In monitoring service quality or time to an event, among others, decreasing process median or variability are usually considered as a process improvement and not something to worry about in practice. We only desire to have a quick signal when there is an increase in the median or scatter as they refer to a deterioration in the process quality. On the contrary, in network traffic surveillance, a decrease in process variability could be a matter of concern as it may result in network congestion where a decreasing shift may be of more interest. In such instances, specific one-sided shifts are of more importance, and a one-sided SDFS scheme would be more appropriate than traditional two-sided schemes. In this article, we propose several Shewhart-Cucconi type SDFS schemes for jointly monitoring one-sided deviations in location or scale or both parameters of a process. We discuss the implementation steps and establish the in-control robustness of the new surveillance plans. We compare proposed schemes with existing one-sided Shewhart-Lepage type schemes using run-length properties. Numerical results substantiate that the proposed schemes perform quite well in many cases. We explain the application of newly designed schemes in surveillance of waiting time during a taxi ride in some cities.

Attractors associated to a family of hyperbolic -adic plane automorphisms

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.

Subshifts, λ-graph bisystems and C⁎-algebras

We introduce a notion of λ-graph bisystem that consists of a pair (L−,L+) of two labeled Bratteli diagrams L−,L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale groupoids constructed from (L−,L+). We study structure of the C⁎-algebras, and show that they are universal unital unique C⁎-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.

Topological conjugacy of topological Markov shifts and Ruelle algebras

We will characterize topological conjugation for two-sided topological Markov shifts (¯¯¯¯¯XA,¯¯¯σA) in terms of the associated asymptotic Ruelle C∗-algebra RA and its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical circle action. We will also show that the extended Ruelle algebra ˜RA, which is a unital and purely infinite version of RA, together with its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical torus action γA is a complete invariant for topological conjugacy of (¯¯¯¯¯XA,¯¯¯σA). The diagonal action of γA has a unique KMS-state on ˜RA, which is an extension of the Parry measure on ¯¯¯¯¯XA.

Some simplified Shewhart-type distribution-free joint monitoring schemes and its application in monitoring drinking water turbidity

In this article, we propose three nonparametric schemes based on the combination of p-values for simultaneous monitoring of shifts in location and scale parameters of a process. Li and Qiu first introduced the idea of dynamic nonparametric monitoring based on the p-value in the context of location shift of a process. This article extends their idea to a more general problem of joint monitoring of location and scale. In this article, we use three approaches for combining the p-values, namely Fisher, Liptak, and Tippett methods, to design three new schemes. We refer to them as the SL-Fisher (hereafter, SLF), the SL-Liptak (hereafter, SLL), and the SL-Tippett (hereafter, SLT) schemes. These schemes are based on a single plotting statistic attained by combining the p-values of the Wilcoxon-rank-sum (WRS) and Ansari-Bradley (AB) test statistics along with the combining functions to control the multiplicity effect. We use a real application based on water turbidity data to explain the implementation of the proposed schemes. We examine and compare the performance characteristics including the mean, median, and several percentiles of the proposed schemes thoroughly using Monte-Carlo simulation. We observe that, in general, the SLT scheme is effective for the detection of shifts in scale followed by either no shift or small shift in the location and the SLL scheme for other cases of shifts. Unlike existing schemes, the proposed schemes offer easier post-signal follow-up approach. We illustrate the design for one-sided shifts in both location and scale, but it can be easily extended for two sided shifts in both location and scale or two-sided shifts in location along with one-sided shifts in scale.

Smale Horseshoe in a Piecewise Smooth Map

We investigate the chaotic dynamics of a two-dimensional piecewise smooth map. The map represents the normal form of a discrete time representation of impact oscillators near grazing states. It is proved that, in certain region of the parameter space, the nonwandering set of the map is contained in a bounded region and that, restricted to the nonwandering set, the map is topologically conjugate to the two-sided shift map on two symbols.

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C⁎-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

Characterizations of Exponential Distribution Based on Two-Sided Random Shifts

A new characterization of the exponential distribution is obtained. It is based on an equation involving randomly shifted (translated) order statistics. No specific distribution is assumed for the shift random variables. The proof uses a recently developed technique including the Maclaurin series expansion of the probability density of the parent variable.

On locally compact shift-continuous topologies on the α-bicyclic monoid

Abstract A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x ↦ axb, is continuous. For every ordinal α ≤ ω, we describe all shift-continuous locally compact Hausdorff topologies on the α-bicyclic monoid Bα. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on Bα is anti-isomorphic to the segment of [1, α] of ordinals, endowed with the natural well-order. Also we prove that for each ordinal α the α + 1-bicyclic monoid Bα+1 is isomorphic to the Bruck extension of the α-bicyclic monoid Bα.

Non-Archimedean Hénon maps, attractors, and horseshoes

We study the dynamics of the Henon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Henon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Henon maps over $${\mathbb {Q}}_3$$ , this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over $${\mathbb {Q}}_5$$ and $${\mathbb {Q}}_7$$ as well. On a different region of the parameter space, we show that the Henon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols.

Some distribution-free Lepage-type schemes for simultaneous monitoring of one-sided shifts in location and scale

Abstract From monitoring network traffic to health care systems, product quality to customer satisfaction, system reliability to certain economic parameters, Statistical Process Monitoring (SPM) schemes are widely used as an important statistical tool. Many of the complex processes have unknown or complicated distributions, where distribution-free process monitoring techniques are recommended. In the present world, it is important to monitor both the process location, for example, median, and the process scale as a measure of variability. In recent years, several distribution-free process monitoring schemes are introduced in the literature to monitor both the location and scale parameters simultaneously. Interestingly, however, all these schemes are intended for two-sided shifts in location-scale. More often, only one-sided shifts are important. For example, in a manufacturing industry, usually the reduction in variation of forged piston rings is welcome and we only need to monitor an increase in variation. On the other hand, in the context of network traffic, a low variation may lead to network congestion and needs to be monitored. To this end, we introduce a class of Shewhart-Lepage type schemes for monitoring one-sided shifts. We also investigate performances of the various schemes and compare them in terms of run length properties, such as mean, standard deviation, median and some percentiles. We carry out large scale Monte-Carlo simulation and our computational findings are extremely encouraging. We illustrate the implementation of the proposed chart using two sets of real data.

TWO-SIDED SHIFT SPACES OVER INFINITE ALPHABETS

Ott, Tomforde and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea, we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets, our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any$M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.

Phase II synthetic exponential charts and effect of parameter estimation

Exponential charts for monitoring time-between-events data are widely studied recently as it has shown a wide range in different applications. To improve the sensitivity of the simple exponential chart, this paper develops a synthcontrol chart to monitor the two-sided shifts in the exponential mean. This synthetic chart consists of an exponential sub-chart and a conforming run length (CRL) sub-chart and it is designed from two new perspectives. Previous synthetic charts are constructed based on the optimization program for a specified shift size, which proves to be difficult to predict in practice. In this paper, the synthetic chart is constructed based on a new design method which takes the mean (ARL) and the standard deviation (SDRL) of the run length into consideration. The other improvement is that the exponential sub-chart is designed to ensure that the CRL sub-chart only needs a lower control limit and that the proposed synthetic chart is ARL-unbiased. The effect of parameter estimation on the proposed synthetic chart is also studied in this paper. Performance comparisons show that our proposed chart has a better performance in general.

Control charts for the coefficient of variation

The coefficient of variation (CV) has received an extensive attention in quality control. The existing control charts for monitoring the two-sided changes in the CV do not consider the average run length (ARL) unbiased criterion. In practice, the practitioners prefer the ARL-unbiased control charts. This paper proposes an ARL-unbiased Shewhart-type control chart to monitor the two-sided shifts in the CV. Then an ARL-unbiased synthetic chart in which the conforming run length sub-chart only needs a lower control limit is proposed to improve the performance of the ARL-unbiased Shewhart-type chart. The comparison results show that the proposed synthetic chart outperforms the proposed ARL-unbiased control chart in terms of the ARL. The effects of the parameter estimation on the proposed charts are also studied. Finally, a real example is used to illustrate the proposed control charts.