Minimal Cantor Systems 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.

Topological speedups for minimal Cantor systems

Topological speedups for minimal Cantor systems

Local Trivializations of Suspended Minimal Cantor Systems and the Stable Orbit-Breaking Subalgebra

It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting \(C^*\)-algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice. This construction provides an alternative characterization (up to stabilization) of the orbit-breaking subalgebra introduced by Putnam for minimal homeomorphisms of Cantor spaces in terms of suspension flows associated to such dynamical systems.

Realizing semicomputable simplices by computable dynamical systems

We study the computability of the set of invariant measures of a computable dynamical system. It is known to be semicomputable but not computable in general, and we investigate which semicomputable simplices can be realized in this way. We prove that every semicomputable finite-dimensional simplex can be realized, and that every semicomputable finite-dimensional convex set is the projection of the set of invariant measures of a computable dynamical system. In particular, there exists a computable system having exactly two ergodic measures, none of which is computable. Moreover, all the dynamical systems that we build are minimal Cantor systems.

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].

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.

On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper $${\mathcal {S}}$$ S -adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux–Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of balancedness for primitive unimodular proper S-adic subshifts.

All minimal Cantor systems are slow

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

Eigenvalues of minimal Cantor systems

In this article we give necessary and sufficient conditions for a complex number to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure-theoretical eigenvalue. These conditions are established from the combinatorial information on the Bratteli–Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational continuous eigenvalues which shares all measure-theoretical eigenvalues with the original system. In a second application a minimal Cantor system is constructed satisfying the so-called maximal continuous eigenvalue group property.

Self-induced systems

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous or expansive case. Nevertheless, we exhibit a zero entropy self-induced system that is neither equicontinuous nor expansive. We also provide non-uniquely ergodic self-induced systems with infinite entropy. Moreover, we give a characterization of self-induced minimal Cantor systems in terms of substitutions on finite or infinite alphabets.

Bounded topological speedups

ABSTRACTThis paper explores the range of bounded speedups in the topological category. Bounded speedups represent both a strengthening of topological speedups as first defined by the second author. and a generalization of powers of a transformation. Here we show that bounded speedups preserve the structure of two classical minimal Cantor systems. Specifically, a minimal bounded speedup of an odometer is a conjugate odometer, and a minimal bounded speedup of a primitive substitution is again a primitive substitution, though it is never conjugate to the original substitution system. Further, we give bounds on the topological entropy of bounded speedups, and in special cases, we compute the topological entropy of bounded speedups.

Eigenvalues and strong orbit equivalence

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.

Linking invariants for smooth minimal solenoids

In this paper we consider smooth diffeomorphisms of the 2-disk which are the identity on the boundary. We assume that the dynamics of any such a diffeomorphism φ restricted to a φ-invariant Cantor set is minimal and uniquely ergodic. Then the average linking number of the orbits of φ can be computed in two standard ways. We prove that the asymptotic average of the diagonal component of the Calabi invariant coincides with the Ruelle invariant of the minimal Cantor system.

Formula omitted]-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β:Sn→Sn, for n=2k+1, k≥1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (Sn,β) there is a Cantor minimal system (X,α) such that the corresponding product system (X×Sn,α×β) is minimal and the resulting crossed product C⁎-algebra C(X×Sn)⋊α×βZ is tracially approximately an interval algebra (TAI). This entails classification for such C⁎-algebras. Moreover, the minimal Cantor system (X,α) is such that each tracial state on C(X×Sn)⋊α×βZ induces the same state on the K0-group and such that the embedding of C(Sn)⋊βZ into C(X×Sn)⋊α×βZ preserves the tracial state space. This implies C(Sn)⋊βZ is TAI after tensoring with the universal UHF algebra, which in turn shows that the C⁎-algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.

Graph covers and ergodicity for zero-dimensional systems

Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.

Amenable minimal Cantor systems of free groups arising from diagonal actions

Abstract We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

Topological orbit equivalence classes and numeration scales of logistic maps

AbstractWe show that every uniquely ergodic minimal Cantor system is topologically orbit equivalent to the natural extension of a numeration scale associated to a logistic map.

Invariant measures of minimal post-critical sets of logistic maps

We construct logistic maps whose restriction to the ω-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the ω-limit set of its critical point. Furthermore, we show that such a logistic map f can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of f for the potential −ln |f′|.

On the eigenvalues of finite rank Bratteli–Vershik dynamical systems

AbstractIn this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

Minimal Cantor Systems and Unimodal Maps

We consider the restriction of unimodal maps f to the omega-limit set y ( c ) of the critical point for certain cases where y ( c ) is a Minimal Cantor set. We investigate the relation of these minimal systems to enumeration scales (generalized adding machines), to Vershik adic transformations on ordered Bratelli diagrams and to substitution shifts. Sufficient conditions are given for ( y ( c ), f ) to be uniquely ergodic.