Topological Rank Research Articles

Subshifts of finite symbolic rank

Abstract The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank- $1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$ , and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra A(p) of entire functions are established. For a given fixed sequence (p(n))n≥0 of positive real numbers, such that limn→∞⁡p(n)1n=∞, the Banach algebra A(p) is the set of all entire functions f such that f(z)=∑n=0∞fˆ(n)zn (z∈C), where the sequence (fˆ(n))n≥0 of Taylor coefficients of f satisfies fˆ(n)=O(p(n)−1) for n→∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication ⁎ with weight given by p (i.e., (f⁎g)(z)=∑n=0∞p(n)fˆ(n)gˆ(n)zn for all z∈C), and the norm ‖f‖=supn≥0⁡p(n)|fˆ(n)|. The following results are shown:•The Topological stable rank of A(p) is 1.•The Bass stable rank of A(p) is 1.•A(p) is a Hermite ring.•A(p) is not a projective-free ring.•Idempotents in A(p) are described.•Exponentials in A(p) are described, and it is shown that every invertible element of A(p) has a logarithm, so that the first Čech cohomology group H1(M(A(p)),Z) with integer coefficients of the maximal ideal space M(A(p)) is trivial.•A generalised necessary and sufficient ‘corona-type condition’ on the matricial data (A,b) with entries from A(p) is given for the solvability of Ax=b with x also having entries from A(p).•The Krull dimension of A(p) is infinite.•A(p) is neither Artinian nor Noetherian.•A(p) is coherent.•The special linear group over A(p) is generated by elementary matrices.

The Frattini subgroup of a Lie group and the topological rank of a Lie group

We define the Frattini subgroup of a topological group G as the set of non-generators. We describe it completely for connected Lie groups, first for the case that G is solvable and then in general. As an application we determine the topological rank of G, that is the minimal number of elements needed to generate a dense subgroup of G.

Dynamical properties of minimal Ferenczi subshifts

AbstractWe provide an explicit $\mathcal {S}$ -adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.

Symbolic factors of -adic subshifts of finite alphabet rank

AbstractThis paper studies several aspects of symbolic (i.e. subshift) factors of $\mathcal {S}$ -adic subshifts of finite alphabet rank. First, we address a problem raised by Donoso et al [Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity. Trans. Amer. Math. Soc.374(5) (2021), 3453–3489] about the topological rank of symbolic factors of $\mathcal {S}$ -adic subshifts and prove that this rank is at most the one of the extension system, improving on the previous results [B. Espinoza. On symbolic factors of S-adic subshifts of finite alphabet rank. Preprint, 2022, arXiv:2008.13689v2; N. Golestani and M. Hosseini. On topological rank of factors of Cantor minimal systems. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2021.62. Published online 8 June 2021]. As a consequence of our methods, we prove that finite topological rank systems are coalescent. Second, we investigate the structure of fibers $\pi ^{-1}(y)$ of factor maps $\pi \colon (X,T)\to (Y,S)$ between minimal ${\mathcal S}$ -adic subshifts of finite alphabet rank and show that they have the same finite cardinality for all y in a residual subset of Y. Finally, we prove that the number of symbolic factors (up to conjugacy) of a fixed subshift of finite topological rank is finite, thus extending Durand’s similar theorem on linearly recurrent subshifts [F. Durand. Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. & Dynam. Sys.20(4) (2000), 1061–1078].

On a measurable analogue of small topological full groups II

We pursue the study of L 1 full groups of graphings and of the closures of their derived groups, which we call derived L 1 full groups. Our main result shows that aperiodic probability measure-preserving actions of finitely generated groups have finite Rokhlin entropy if and only if their derived L 1 full group has finite topological rank. We further show that a graphing is amenable if and only if its L 1 full group is, and explain why various examples of (derived) L 1 full groups fit very well into Rosendal’s geometric framework for Polish groups. As an application, we obtain that every abstract group isomorphism between L 1 full groups of amenable ergodic graphings must be a quasi-isometry for their respective L 1 metrics. We finally show that L 1 full groups of rank one transformations have topological rank 2.

Topological rank of (-1, 1) metabelian algebras

In 1981, Pchelintsev developed the idea for arranging non-nilpotent subvarieties in a given variety by using topological rank for spechtian varieties of algebra as a fixed tool. In this paper we show that for a given topological rank over a field of 2, 3 ? torsion free of (-1; 1) metabelian algebra solvable of index 2 that are Lie-nilpotent of step not more than p is equal to P.

The profinite topology of free groups and weakly generic tuples of automorphisms

AbstractLet be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number . Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures.

On topological rank of factors of Cantor minimal systems

AbstractA Cantor minimal system is of finite topological rank if it has a Bratteli–Vershik representation whose number of vertices per level is uniformly bounded. We prove that if the topological rank of a minimal dynamical system on a Cantor set is finite, then all its minimal Cantor factors have finite topological rank as well. This gives an affirmative answer to a question posed by Donoso, Durand, Maass, and Petite in full generality. As a consequence, we obtain the dichotomy of Downarowicz and Maass for Cantor factors of finite-rank Cantor minimal systems: they are either odometers or subshifts.

Homological mirror symmetry without correction

Let X X be a closed symplectic manifold equipped with a Lagrangian torus fibration over a base Q Q . A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space Y Y , which can be considered as a variant of the T T -dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of tautologically unobstructed graded Lagrangians in X X embeds fully faithfully in the derived category of (twisted) coherent sheaves on Y Y , under the technical assumption that π 2 ( Q ) \pi _2(Q) vanishes (all known examples satisfy this assumption). The main new tool is the construction and computation of Floer cohomology groups of Lagrangian fibres equipped with topological infinite rank local systems that correspond, under mirror symmetry, to the affinoid rings introduced by Tate, equipped with their natural topologies as Banach algebras.

Interplay between finite topological rank minimal Cantor systems, 𝒮-adic subshifts and their complexity

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable S {\mathcal S} -adic subshifts. This is done by establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like many classical zero-entropy examples) have finite topological rank. Conversely, we analyze the complexity of S {\mathcal S} -adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so-called left to right S {\mathcal S} -adic subshifts. We also show that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank two subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.

Tracial topological rank zero and stable rank one for certain tracial approximation C∗-algebras

Tracial topological rank zero and stable rank one for certain tracial approximation C∗-algebras

Regularity Properties of Certain Crossed Product C*-Algebras

We show that the following properties of C* -algebras in a class Ω are inherited by simple unital C*-algebras in class TAΩ: (1) m-comparison of positive elements, (2) strong tracial m-comparison of positive elements, and (3) tracial nuclear dimension at most m. As an application, every unital simple C*-algebra with tracial topological rank at most k has tracial nuclear dimension at most k. Also as an application, let A be an infinite-dimensional simple unital exact C*-algebra such that A has one of the above-listed properties. Suppose that α: G → Aut(A) is an action of a finite group G on A which has the tracial Rokhlin property. Then the crossed product C*-algebra C* (G, A, α) also has the property under consider also.

Note on compact operators on a Hilbert C⁎-module

In this note, we characterize the tracial topological rank of the ideal of compact operators on a Hilbert A-module. It is also proved that this ideal preserves exactness, quasidiagonality, and nuclearity if the C⁎-algebra A has these properties, respectively. Finally, a necessary and sufficient condition is given for this ideal K(E) being an AT-algebra when E is a Hilbert A-module over an AT-algebra A.

Topological stable rank of mathcal {E}'(mathbb {R})

The set mathcal {E}'(mathbb {R}) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of mathcal {E}'(mathbb {R}) is 2.

Comparing the topological rank of journals in Web of Science and Mendeley

Recently, there has been a surge of interest in new data emerged due to the rapid development of the information technologies in scholarly communication. Since the 2010s, altmetrics has become a common trend in scientometric research. However, researchers have not treated in much detail the question of the probability distributions underlying these new data. The principal objective of this study was to investigate one of the classic problems of scientometrics—the problem of citation and readership distributions. The study is based on the data obtained from two information systems: Web of Science and Mendeley. Here we based on the concept of the cumulative empirical distribution function to explore the differences and similarities between citations and readership counts of biological journals indexed in Web of Science and Mendeley. The basic idea was to determine, for any journal, a “size” (it is said to be the topological rank) of citation and readership empirical cumulative distributions, and then to compare distributions of the topological ranks of Web of Science and Mendeley. In order to verify our model, we employ it to the bibliometric and altmetric research of 305 biological journals indexed in Journal Citation Reports 2015. The findings show that both distributions of the topological rank of biological journals are statistically close to the Wakeby distribution. The findings presented in this study add to our understanding of information processes of the scholarly communication in the new digital environment.

Automorphism groups of finite topological rank

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an ω \omega -categorical structure can go down to reducts. Together, those results prove that a large number of ω \omega -categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any ω \omega -categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.

Pathways of topological rank analysis (PoTRA): a novel method to detect pathways involved in hepatocellular carcinoma.

Complex diseases such as cancer are usually the result of a combination of environmental factors and one or several biological pathways consisting of sets of genes. Each biological pathway exerts its function by delivering signaling through the gene network. Theoretically, a pathway is supposed to have a robust topological structure under normal physiological conditions. However, the pathway’s topological structure could be altered under some pathological condition. It is well known that a normal biological network includes a small number of well-connected hub nodes and a large number of nodes that are non-hubs. In addition, it is reported that the loss of connectivity is a common topological trait of cancer networks, which is an assumption of our method. Hence, from normal to cancer, the process of the network losing connectivity might be the process of disrupting the structure of the network, namely, the number of hub genes might be altered in cancer compared to that in normal or the distribution of topological ranks of genes might be altered. Based on this, we propose a new PageRank-based method called Pathways of Topological Rank Analysis (PoTRA) to detect pathways involved in cancer. We use PageRank to measure the relative topological ranks of genes in each biological pathway, then select hub genes for each pathway, and use Fisher’s exact test to test if the number of hub genes in each pathway is altered from normal to cancer. Alternatively, if the distribution of topological ranks of gene in a pathway is altered between normal and cancer, this pathway might also be involved in cancer. Hence, we use the Kolmogorov–Smirnov test to detect pathways that have an altered distribution of topological ranks of genes between two phenotypes. We apply PoTRA to study hepatocellular carcinoma (HCC) and several subtypes of HCC. Very interestingly, we discover that all significant pathways in HCC are cancer-associated generally, while several significant pathways in subtypes of HCC are HCC subtype-associated specifically. In conclusion, PoTRA is a new approach to explore and discover pathways involved in cancer. PoTRA can be used as a complement to other existing methods to broaden our understanding of the biological mechanisms behind cancer at the system-level.

Orbit full groups for locally compact groups

We show that the topological rank of an orbit full group generated by an ergodic, probability measure-preserving free action of a non-discrete unimodular locally compact Polish group is two. For this, we use the existence of a cross section and show that for a locally compact Polish group, the full group generated by any dense subgroup is dense in the orbit full group of the action of the group. We prove that the orbit full group of a free action of a locally compact Polish group is extremely amenable if and only if the acting group is amenable, using the fact that the full group generates the von Neumann algebra of the action.

When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ1(Z) nor in ℓ1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.