Subshift Research Articles

MIMO GS OVSF/OFDM Based Underwater Acoustic Multimedia Communication Scheme

An underwater acoustic multimedia communication (UWAMC) system is proposed with 2400 transmission modes according to time-varying multipath underwater acoustic (UWA) channel conditions. The orthogonal variable spreading factor (OVSF) scheme and Gold sequence (GS) scramble code are integrated into multi-input multi-output UWAMC system based on orthogonal frequency-division multiplexing to achieve the quality of service of multimedia transmission in the UWA channel. Binary phase shift keying (BPSK) and quadrature phase shift keying (QPSK) adaptive modulation, direct mapping (DM) or space–time block code (STBC) transmission strategies, convolution channel code with rate 1/2 and 1/3, and a power assignment mechanism were adopted in the proposed system. Simulation results show that the bit error rate (BER) and power saving ratio (PSR) performance of the STBC strategy with transmission diversity is superior to that of the DM strategy without transmission diversity, and the performance of the BERs and PSRs of the transmission scheme with the GS scramble code surpasses that of the scheme without the code. The performance of the BERs and PSRs of BPSK modulation with a channel code rate of 1/3 is better than that of BPSK modulation with a channel code rate of 1/2, and the performances of BERs and PSRs of BPSK modulation with a channel code rate of 1/3 are better than that of QPSK modulation with a channel code rate of 1/3. As the length of the OVSF codes increases, the UWAMC system’s BERs decrease, and its PSRs increase. The UWAMC system can achieve either maximum transmission speed or maximum transmission power efficiency.

Arcwise connectedness of the set of ergodic measures of hereditary shifts

We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology, and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok.

Entropy of Subordinate Shift Spaces

We introduce a new family of shift spaces—the subordinate shifts. Using subordinate shifts, we prove in an elementary way that for every nonnegative real number t there is a shift space with entropy t.

Mixing properties for hom-shifts and the distance between walks on associated graphs

Let $\mathcal H$ be a finite connected undirected graph and $\mathcal H_{walk}$ be the graph of bi-infinite walks on $\mathcal H$; two such walks $\{x_i\}_{i\in \mathbb Z}$ and $\{y_i\}_{i \in \mathbb Z}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i \in \mathbb Z$. We consider the question: Given a graph $\mathcal H$ when is the diameter (with respect to the graph metric) of $\mathcal H_{walk}$ finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of $\mathbb Z^d$ with respect to the standard generators to $\mathcal H$) and are the subject of this paper.

Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces

Sharp estimates for mean square approximations of classes of differentiable periodic functions by shift spaces

Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let $${{\Bbb S}_{B,n}}$$ be the space of functions s of the form $$s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} $$ . This paper describes all spaces $${{\Bbb S}_{B,n}}$$ that satisfy the exact inequality $$E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}$$ . (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

Reversing and extended symmetries of shift spaces

The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue–Morse system with various generalisations or the Rudin–Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries . We develop their basic theory for faithful \begin{document}$\mathbb{Z}^{d}$\end{document} -actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.

A necessary and sufficient condition for a matrix equilibrium state to be mixing

Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved by Feng and Käenmäki [Equilibrium states of the pressure function for products of matrices.Discrete Contin. Dyn. Syst.30(3) (2011), 699–708]. In a recent article [Ergodic properties of matrix equilibrium states.Ergod. Th. & Dynam. Sys.(2017), to appear] the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties, but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.

Mixing properties of tree-shifts

The uncertainty of multidimensional shift spaces draws attracted attention of researchers. For example, the emptiness problem is undecidable; there exist aperiodic shifts of finite type; there is a nonempty shift of finite type exhibiting nonextensible local patterns. This paper investigates symbolic dynamics on Cayley trees and gives affirmative answers to the above questions in tree-shifts. Beyond that, with introducing block gluing tree-shift, a sufficient condition of exhibiting positive topological entropy is revealed.

Phase transitions in long-range Ising models and an optimal condition for factors of -measures

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

Infinite Alphabet Edge Shift Spaces via Ultragraphs and Their C*-Algebras

We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.

Inverse semigroup shifts over countable alphabets

In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift spaces whose correspondent inverse semigroup operation is a 1-block operation, that is, it arises from a group operation on the alphabet. Motivated by this, we go on to study block operations on shift spaces and, in the end, we prove our main theorem, which states that Markovian shift spaces, which can be endowed with a 1-block inverse semigroup operation, are conjugate to the product of a full shift with a fractal shift.

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

In this paper, we will study representations of the continuous full group \Gamma_A of a one-sided topological Markov shift (X_A,\sigma_A) for an irreducible matrix A with entries in \{0,1\} as a generalization of Higman–Thompson groups V_N, 1 < N \in {\mathbb{N}} . We will show that the group \Gamma_A can be represented as a group \Gamma_A^{\operatorname{tab}} of matrices, called A -adic tables, with entries in admissible words of the shift space X_A , and a group \Gamma_A^{\operatorname{PL}} of right continuous piecewise linear functions, called A -adic PL functions, on [0,1] with finite singularities.

On automorphism groups of Toeplitz subshifts

On automorphism groups of Toeplitz subshifts, Discrete Analysis 2017:11, 19 pp. A discrete dynamical system is a space $X$ with some kind of structure, together with a map $\sigma\colon X\to X$ that preserves the structure. (For instance, if $X$ is a topological space, then one asks for $\sigma$ to be continuous, and if $X$ is a differentiable manifold, then one asks for it to be a diffeomorphism.) Given such a system, one studies the structure of the orbits $x, \sigma x, \sigma^2x, \dots$ that are obtained by iterating the map $\sigma$. A particularly interesting subfield of dynamics is _symbolic dynamics_, where $X$ is a space of bi-infinite sequences over a finite alphabet $A$, $X$ is closed under the left shift, and $\sigma$ is that left shift. One also asks for $X$ to be closed in the topological sense: we take the discrete topology on $A$ and the product topology on $A^{\mathbb Z}$, of which $X$ is a subset. A system $(X,\sigma)$ is called a _shift space_. Such spaces can encode interesting combinatorial information, and that has led to a very fruitful interplay between combinatorics and dynamical systems. An _automorphism_ of the system $(X, \sigma)$ is a homeomorphism $\phi\colon X\to X$ that commutes with $\sigma$, and ${\rm Aut}(X, \sigma)$ denotes the group (under composition) of automorphisms of the system. The _complexity_ of a shift system ${\rm Aut}(X, \sigma)$ is the map $p\colon\mathbb N\to \mathbb N$ that counts the number of blocks of length $n$ appearing in the sequences $x\in X$. If the complexity is linear, then the automorphism group is understood for any shift ${\rm Aut}(X, \sigma)$, but beyond linear, the problem becomes complicated. For example, under mild assumptions on the shift ${\rm Aut}(X, \sigma)$, the automorphism group is not finitely generated and it contains isomorphic copies of all finite groups, countably many copies of $\mathbb Z$, and the free group on any finite number of generators. Thus while ${\rm Aut}(X, \sigma)$ is always countable, in general it can be quite complicated and difficult to compute. However, for several reasons it is desirable to do so: for example, it gives a useful invariant. This paper continues recent work on automorphism groups for various classes of shift spaces, computing the automorphism group for the class of Toeplitz shifts, a large class of shift systems frequently used to provide counterexamples in symbolic dynamics. A sequence $x\in A^{\mathbb Z}$ is _Toeplitz_ if every finite block in $x$ appears periodically, and a shift space $(X, \sigma)$ is a _Toeplitz shift_ if $X$ is the orbit closure of some Toeplitz sequence. (It is not hard to construct Toeplitz sequences that are not periodic. For one example, take $x_n$ to be the parity of $k$, where $k$ is maximal such that $2^k|n$.) This rigid structure on $X$ implies that ${\rm Aut}(X, \sigma)$ is Abelian, and this is the starting point for the classification of the automorphism groups of Toeplitz shifts. The authors start with Toeplitz shifts of subquadratic complexity, showing that the automorphism group is spanned by the roots of the shift map $\sigma$ modulo the torsion subgroup $T$ of ${\rm Aut}(X, \sigma)$. More generally, they show that if ${\rm Aut}(X, \sigma)/\langle\sigma\rangle$ is a periodic group, then the automorphism group is spanned by $T$ and the roots of the shift $\sigma$ (that is, the automorphisms $\phi$ such that $\phi^n=\sigma$ for some $n$). Under the further assumption that $T$ is trivial, they show that the automorphism group is either infinite cyclic or is not finitely generated. This method leads to examples of Toeplitz shifts whose complexity is arbitrarily close to linear, in the sense that for every $\varepsilon > 0$ the complexity satisfies the inequality $p(n)\leq Cn^{1+\varepsilon}$ for some constant $C=C_\varepsilon > 0$, such that the automorphism group is not finitely generated. Note that this result cannot be extended to linear complexity, where it is known that the automorphism group is always finitely generated. In the opposite regime, that of high complexity, the authors show that the automorphism group need not be large. Given any infinite and finitely generated Abelian group $G$ with cyclic torsion, they construct a Toeplitz shift with positive entropy (meaning that the complexity function grows exponentially) whose automorphism group is exactly $G$.

Uniform scaling limits for ergodic measures

We provide an elementary proof that ergodic measures on one-sided shift spaces are ‘uniformly scaling’ in the following sense: at almost every point the scenery distributions weakly converge to a common distribution on the space of measures. Moreover, we show how the limiting distribution can be expressed in terms of, and derived from, a 'reverse Jacobian’ function associated with the corresponding measure on the space of left infinite sequences. Finally we specialise to the setting of Gibbs measures, discuss some statistical properties, and prove a Central Limit Theorem for ergodic Markov measures.

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.

Spectral properties of the Ruelle operator for product-type potentials on shift spaces

We study a class of potentials $f$ on one sided full shift spaces over finite or countable alphabets, called potentials of product type. We obtain explicit formulae for the leading eigenvalue, the eigenfunction (which may be discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness property of these quantities is also discussed and it is shown that there always exists a Bernoulli equilibrium state even if $f$ does not satisfy Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N} \to \mathbb{R}$ of the form $$ f(x_1,x_2,\ldots) = x_1 + 2^{-\gamma} \, x_2 + 3^{-\gamma} \, x_3 + ...+n^{-\gamma} \, x_n + \ldots $$ with $\gamma >1$. For $3/2 < \gamma \leq 2$, we obtain the existence of two different eigenfunctions. Both functions are (locally) unbounded and exist a.s. (but not everywhere) with respect to the eigenmeasure and the measure of maximal entropy, respectively.

Robust Head-Pose Estimation Based on Partially-Latent Mixture of Linear Regressions.

Head-pose estimation has many applications, such as social event analysis, human-robot and human-computer interaction, driving assistance, and so forth. Head-pose estimation is challenging, because it must cope with changing illumination conditions, variabilities in face orientation and in appearance, partial occlusions of facial landmarks, as well as bounding-box-to-face alignment errors. We propose to use a mixture of linear regressions with partially-latent output. This regression method learns to map high-dimensional feature vectors (extracted from bounding boxes of faces) onto the joint space of head-pose angles and bounding-box shifts, such that they are robustly predicted in the presence of unobservable phenomena. We describe in detail the mapping method that combines the merits of unsupervised manifold learning techniques and of mixtures of regressions. We validate our method with three publicly available data sets and we thoroughly benchmark four variants of the proposed algorithm with several state-of-the-art head-pose estimation methods.

Uniform global attractors for non-autonomous dissipative dynamical systems

In this paper we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.

The Dynamics of 1-Step Shifts of Finite Type Over Two Symbols

A 1-step shift of finite type over two symbols is a collection of sequences over symbols 0 and 1 with some constrains. The constrains are identified by a set of forbidden blocks which are not allowed to appear in any sequences in the space. The space is of finite type since the number of forbidden blocks is finite and it is of 1-step type since the forbidden blocks are of length of 2. The aim of this paper is to look at the chaotic behaviour of 1-step shift of finite type by considering all spaces of it type. We found that there are six different 1-step shifts of finite type which exhibits totally different dynamics behaviour. We explain the dynamics of each space and then discuss on the difference of the dynamic properties between these spaces. Two of them are chaotic in the sense of Devaney. However the two chaotic shift spaces have totally different behaviour where one of them has trivial dynamics. The other four spaces are not chaotic but, they have some interesting behaviour to be highlighted. It turns out that some of the non-chaotic shift spaces satisfy some chaotic properties.