One-sided Shift Research Articles

On the prevalence of the periodicity of maximizing measures

For a continuous map T:X→X on a compact metric space (X,d), we say that a function f:X→R has the property PT if its time averages along forward orbits of T are maximized at a periodic orbit. In this paper, we prove that for the one-sided full shift on two symbols, the property PT is prevalent (in the sense of Hunt–Sauer–Yorke) in spaces of Lipschitz functions with respect to metrics with mildly fast decaying rate on the diameters of cylinder sets. This result is a strengthening of [3, Theorem A], confirms the prediction mentioned in the ICM proceeding contribution of J. Bochi ([1, Section 1]) suggested by experimental evidence, and is another step towards the Hunt–Ott conjectures in the area of ergodic optimization.

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.

Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.

Triple Homogeneously Weighted Moving Average Charts for Monitoring Process Dispersion

Homogeneously weighted moving average (HWMA) charts have recently achieved popularity for monitoring small changes in process parameters (location and/or dispersion). Furthermore, the DHWMA (double HWMA) and THWMA (triple HWMA) are the advanced versions of the HWMA charts. The HWMA chart for the process dispersion is designed to detect only the upward (one‐sided) shift (i.e., process deterioration). Employing a two‐sided chart for concurrently detecting both process improvement and process deterioration is an important aspect of statistical process monitoring. By taking this point as motivation, one and two‐sided THWMA charts (symbolized as the THWMAV) are proposed for monitoring the process dispersion. The Monte Carlo simulations are performed to investigate the performance behavior of the THWMAV charts in terms of certain performance indicators, including ARL, SDRL, EQL, RARL, and PCI. The comparison among the THWMAV versus existing charts (DHWMAV, HWMAV, TEWMAV, DEWMAV, and EWMAV) indicates that the THWMAV charts outperform the existing charts. Finally, a dataset is also analyzed to illustrate the implementation of the THWMAV charts.

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.

Weak Formulation of the Laplacian on the Full Shift Space

We consider a Laplacian on the one-sided full shift space over a finite symbol set, which is constructed as a renormalized limit of finite difference operators. We propose a weak definition of this Laplacian, analogous to the one in calculus, by choosing test functions as those which have finite energy and vanish on various boundary sets. In the abstract setting of the shift space, the boundary sets are chosen to be the sets on which the finite difference operators are defined. We then define the Neumann derivative of functions on these boundary sets and establish a relation between three important concepts in analysis so far, namely, the Laplacian, the bilinear energy form and the Neumann derivative of a function. As a result, we obtain the Gauss-Green's formula analogous to the one in classical case. We conclude this paper by providing a sufficient condition for the Neumann boundary value problem on the shift space.

Expansive dynamics on profinite groups

A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite sequence of full one-sided group shifts and two finite groups.

Characterization and topological behavior of homomorphism tree-shifts

The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift X and its associated hom tree-shift TX, as well as the equivalence in the sofic shift. On the other hand, we investigate the interrelationship among the comparable mixing properties on tree-shifts as those on multidimensional shift spaces. They include irreducibility, topologically mixing, block gluing, and strong irreducibility, all of which are defined in the spirit of classical multidimensional shift, complete prefix code (CPC), and uniform CPC. In summary, the mixing properties defined in all three manners coincide for TX. Furthermore, an equivalence between irreducibility on TA and irreducibility on XA are seen, and so is one between topologically mixing on TA and mixing property on XA, where XA is the one-sided shift space induced by the matrix A and TA is the associated tree-shift. These equivalences are consistent with the mixing properties on X or XA when viewed as a degenerate tree-shift.

On structure of topological entropy for tree-shift of finite type

This paper deals with the topological entropy for hom Markov shifts T M on d -tree. If M is a reducible adjacency matrix with q irreducible components M 1 , ⋯ , M q , we show that h ( T M ) = max 1 ≤ i ≤ q ⁡ h ( T M i ) fails generally, and present a case study with full characterization in terms of the equality. Though that it is likely the sets { h ( T M ) : M is binary and irreducible } and { h ( T X ) : X is a one-sided shift } are not coincident, we show the two sets share the common closure. Despite the fact that such closure is proved to contain the interval [ d log ⁡ 2 , ∞ ) , numerical experiments suggest its complement contain open intervals.

Distribution of Typical Orbits for Random Dynamics Generated by Finitely Many Rational Maps

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, $$R_{1}, R_{2}, \ldots , R_{N}$$ , each with degree $$d_{i};\ 1 \le i \le N$$ and with at least one $$R_{i}$$ in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanski (equilibrium) measure associated to some real-valued Holder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

Decimation and interleaving operations in one-sided symbolic dynamics

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show these operations are closed under composition. We also study a family of n-ary interleaving operations, one for each n≥1. Given subsets X0,X1,...,Xn−1 of such a shift space, the n-ary interleaving operation produces a set whose elements combine individual elements xi, one from each Xi, by interleaving their symbol sequences in arithmetic progressions (modn). We determine algebraic relations between decimation and interleaving operations and the shift map. We study set-theoretic n-fold closure operations X↦X[n], which interleave decimations of X of modulus level n. A set is n-factorizable if X=X[n]. The n-fold interleaving operations are closed under composition and are idempotent. To each X we assign the set N(X) of all values n≥1 for which X=X[n]. We characterize the possible sets N(X) as nonempty sets of positive integers that form a distributive lattice under the divisibility partial order and are downward closed under divisibility. We show that all sets of this type occur. We introduce a class of weakly shift-stable sets and show that this class is closed under all decimation, interleaving, and shift operations. We study two notions of entropy for subsets of the full one-sided shift and show that they coincide for weakly shift-stable X, but can be different in general. We give a formula for entropy of interleavings of weakly shift-stable sets in terms of individual entropies.

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 $(

𝐶*-algebras, groupoids and covers of shift spaces

To every one-sided shift space X \mathsf {X} we associate a cover X ~ \widetilde {\mathsf {X}} , a groupoid G X \mathcal {G}_\mathsf {X} and a C ∗ \mathrm {C^*} -algebra O X \mathcal {O}_\mathsf {X} . We characterize one-sided conjugacy, eventual conjugacy and (stabilizer-preserving) continuous orbit equivalence between X \mathsf {X} and Y \mathsf {Y} in terms of isomorphism of G X \mathcal {G}_\mathsf {X} and G Y \mathcal {G}_\mathsf {Y} , and diagonal-preserving ∗ ^* -isomorphism of O X \mathcal {O}_\mathsf {X} and O Y \mathcal {O}_\mathsf {Y} . We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces Λ X \Lambda _\mathsf {X} and Λ Y \Lambda _\mathsf {Y} in terms of isomorphism of the stabilized groupoids G X × R \mathcal {G}_\mathsf {X}\times \mathcal {R} and G Y × R \mathcal {G}_\mathsf {Y}\times \mathcal {R} , and diagonal-preserving ∗ ^* -isomorphism of the stabilized C ∗ \mathrm {C^*} -algebras O X ⊗ K \mathcal {O}_\mathsf {X}\otimes \mathbb {K} and O Y ⊗ K \mathcal {O}_\mathsf {Y}\otimes \mathbb {K} . Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of X \mathsf {X} from the pair ( O X , C ( X ) ) (\mathcal {O}_\mathsf {X}, C(\mathsf {X})) , and the flow equivalence class of Λ X \Lambda _\mathsf {X} from the pair ( O X ⊗ K , C ( X ) ⊗ c 0 ) (\mathcal {O}_\mathsf {X}\otimes \mathbb {K}, C(\mathsf {X})\otimes c_0) . In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.

Topological full groups and t.d.l.c. completions of Thompson’s V

We show that every topological full group coming from a one-sided irreducible shift of finite type, in the sense of Matui, is isomorphic to a group of tree almost automorphisms preserving an edge colouring of the tree. As an application, we prove that it admits totally disconnected, locally compact completions of arbitrary nonempty finite local prime content. The completions we construct have open, simple commutator subgroup. Our results apply to Thompson’s V, providing answers to two questions asked by Le Boudec and Wesolek.

Characterization for entropy of shifts of finite type on Cayley trees**This work is partially supported by the Ministry of Science and Technology, ROC (Contract No. MOST 107-2115-M-259 -001 -MY2 and 107-2115-M-390 -002 -MY2).

The notion of tree-shifts constitutes an intermediate class between one-sided shift spaces and multidimensional ones. This paper proposes an algorithm for computing the entropy of a tree-shift of finite type. Meanwhile, the entropy of a tree-shift of finite type is for some , where λ is a Perron number. This extends Lind’s work (1984 Ergod. Theor. Dynam. Syst. 4 283–300) on one-dimensional shifts of finite type. As an application, the entropy minimality problem is investigated, and we obtain a necessary and sufficient condition for a tree-shift of finite type to be entropy-minimal with some additional conditions.

Natural extensions for [formula omitted]-adic [formula omitted]-shifts and other scaling maps

The p-adic β-shift can naturally be thought of as an isometry on the p-adic integers (multiplication by a unit), followed by an iterate of the one-sided shift. We show that all (p−n,pn)-locally scaling maps have such a decomposition into an iterate of the one-sided shift composed with an isometry. We then use this decomposition to give a formula for the natural extension.

CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

AbstractA one-sided shift of finite type$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$determines on the one hand a Cuntz–Krieger algebra${\mathcal{O}}_{A}$with a distinguished abelian subalgebra${\mathcal{D}}_{A}$and a certain completely positive map$\unicode[STIX]{x1D70F}_{A}$on${\mathcal{O}}_{A}$. On the other hand,$(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$determines a groupoid${\mathcal{G}}_{A}$together with a certain homomorphism$\unicode[STIX]{x1D716}_{A}$on${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of$\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

Coloured Neretin groups

We give sufficient conditions for a subgroup of a tree almost automorphism group to be isomorphic to the topological full groups of a one-sided shift in the sense of Matui. As an application,we show that almost automorphismgroups of trees obtained from universal groups constructed by Burger and Mozes are compactly generated and virtually simple. In addition, using the approach of Bader, Caprace, Gelander and Mozes we show that some of these almost automorphism groups do not have any lattice.

The suspension of a graph, and associated C⁎-algebras

Given a directed graph E, we construct for each real number l a quiver whose vertex space is the topological realisation of E, and whose edges are directed paths of length l in the vertex space. These quivers are not topological graphs in the sense of Katsura, nor topological quivers in the sense of Muhly and Tomforde. We prove that when l=1 and E is finite, the infinite-path space of the associated quiver is homeomorphic to the suspension of the one-sided shift of E. We call this quiver the suspension of E. We associate both a Toeplitz algebra and a Cuntz–Krieger algebra to each of the quivers we have constructed, and show that when l=1 the Cuntz–Krieger algebra admits a natural faithful representation on the ℓ2-space of the suspension of the one-sided shift of E. For graphs E in which sufficiently many vertices both emit and receive at least two edges, and for rational values of l, we show that the Toeplitz algebra and the Cuntz–Krieger algebra of the associated quiver are homotopy equivalent to the Toeplitz algebra and Cuntz–Krieger algebra respectively of a graph that can be regarded as encoding the lth higher shift associated to the one-sided shift space of E.

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.