To study any dynamical system, it is useful to find a partition that allows essentially faithful encoding (injective, up to a small exceptional set) into a subshift. Most topological and measure-theoretic systems can be represented by Bratteli–Vershik (or adic, or BV) systems. So, it is natural to ask when can a BV system be encoded essentially faithfully. Extending what was known previously for special diagrams and particular orders, we show here that for BV diagrams defined by homogeneous positive integer multivariable polynomials, and a wide family of their generalizations, which we call polynomial shape diagrams, for every choice of the edge ordering, the coding according to initial path segments of a fixed finite length is injective off of a negligible exceptional set.

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle _{n=0}^\infty $ and an algebraic number $\beta $ such that $|\beta |>1$, the number $[\![ \boldsymbol {u} ]\!] _\beta :=\sum _{n=0}^\infty ({u_n}/{\beta ^n})$ either lies in $\mathbb Q(\beta )$ or is transcendental. In this paper, we show a similar rational–transcendental dichotomy for sequences defined by irreducible Pisot morphisms on binary alphabets. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases, we are able to show transcendence of $[\![ \boldsymbol {u}]\!] _{\beta }$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the k-Bonacci word, then $[\![ \boldsymbol {u}]\!] _{\beta }$ is transcendental.

The density of a rational language can be understood as the frequency of some pattern in the shift space, for example, a pattern like ‘words with an even number of a given letter’. We study the density of group languages, that is, rational languages recognized by morphisms onto finite groups, inside shift spaces. We show that the density with respect to any given ergodic measure on a shift space exists for every group language, because it can be computed by using any ergodic lift of the given measure to a skew product between the shift space and the recognizing group. We then further study densities in shifts of finite type (with a suitable notion of irreducibility) and then in minimal shifts. In the latter case, we obtain a closed formula for the density under the condition that the aforementioned skew product has minimal closed invariant subsets that are ergodic under the product of the original measure and the uniform probability measure on the group. The formula is derived in part from a characterization of minimal closed invariant subsets for skew products between shifts and finite groups relying on notions of cocycles and coboundaries. In the case where the whole skew product is ergodic under the product measure, then the density is just the cardinality of the subset of the group that defines the language divided by the cardinality of the group. Moreover, we provide sufficient conditions for the skew product to have minimal closed invariant subsets that are ergodic under the product measure. Finally, we investigate the link between minimal closed invariant subsets, return words, and bifix codes.

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman–Thompson groups.

We show that the canonical anticommutation relations (CAR) algebra admits a Cantor spectrum $\mathrm {C}^\ast $-diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of $\mathrm {C}^\ast $-algebras, and the diagonal arises by realizing the CAR algebra as the crossed product of a free minimal action on the Cantor space, where the acting group is the product of a locally finite group with the infinite dihedral group. The main ingredient in the construction is a binary subshift associated to the well-known regular paper-folding sequence. Moreover, we show that the CAR algebra in fact admits countably many, pairwise non-conjugate, Cantor spectrum diagonals which are distinguished by the different values of their diagonal dimension, as defined by Li, Liao and the second named author.

We classify the sets of natural numbers n for which certain dynamical systems $(X,f)$ on a compact metric space X have a periodic point of (least) period n. Interest in this question dates back to Sharkovskii’s theorem for continuous maps on intervals of the real line, but it also ties to checkable conditions for Krieger’s embedding theorem for symbolic dynamical systems. Given a system for which the logarithmic derivative of the Artin–Mazur zeta function is rational, we use the Skolem–Mahler–Lech theorem to classify for which n the system has a periodic point of (not necessarily least) period n. Moreover, we build on work on finitely presented (FP) systems and their relationship to symbolic dynamics to classify the set of least periods, that is, periodic orbit lengths, for arbitrary FP systems, extending a known classification for shifts of finite type. We also provide several constructions to realize any such least period sets.

We develop a topological framework for Engel expansions that treats both directions of the correspondence between points of $(0,1]$ and nondecreasing digit sequences. We endow the sequence space with the product topology to study the evaluation map, and we fix a nonterminating digit algorithm to study the digit coding map. We also record the correspondence between cylinder sets and fundamental intervals and give an application to Baire category results for functions of the digits.

The action of a finite group G on a subshift of finite type (SFT) X is called free if every point has trivial stabilizer and it is called inert if the induced action on the dimension group of X is trivial. We show that any two free inert actions of a finite group G on an SFT are conjugate by an automorphism of any sufficiently high power of the shift space. This partially answers a question posed by Fiebig. As a consequence, we obtain that every two free elements of the stabilized automorphism group of a full shift are conjugate in this group. In addition, we generalize a result of Boyle, Carlsen, and Eilers concerning the flow equivalence of G-SFTs.

We study the topological dynamics of the action of an acylindrically hyperbolic group on the space of its infinite index convex cocompact subgroups by conjugation. We show that, for any suitable probability measure $\mu $, random walks with respect to $\mu $ will produce elements with strong mixing properties for this action asymptotically almost surely. In particular, when the group has no finite normal subgroups, this implies that the action is highly topologically transitive. Along the way, we prove technical results about convex cocompact subgroups that allow us to extend some results on random walks of Abbott and the first author.

For every $0 \lt \alpha\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $\alpha$. It has been previously shown by Špitalský that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.

In this paper, we investigate dynamical properties of monoid actions on symbolic systems. First, we establish mixing properties and an entropy formula for such actions. To describe chaotic behavior, we introduce and distinguish several types of Li–Yorke chaos. Our main result shows that positive entropy is equivalent to locally Li–Yorke chaos, a notion that strengthens the classical definition of Li–Yorke chaos.

We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.

We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $\mu $-limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.

In 2002, Kamae and Zamboni [Ergod. Th. & Dynam. Sys. 22(4) (2002), 1191–1199] introduced maximal pattern complexity and determined that any aperiodic sequence must have maximal pattern complexity at least $2k$. In 2006, Kamae and Rao [European J. Combin. 27(1) (2006), 125–137] examined the maximal pattern complexity of sequences over larger alphabets and showed that sequences which have maximal pattern complexity less than $\ell k$, for $\ell $ the size of the alphabet, must have some periodic structure. In this paper, we investigate the structure of sequences of low maximal pattern complexity over $\ell $ letters, where $\liminf _{k \to \infty } p_{\alpha }^*(k) - 3k = -\infty $. In addition, we show that the minimal maximal pattern complexity of an aperiodic sequence which uses all $\ell $ letters is $p_{\alpha }^*(k) = 2k + \ell -2$ and give an exact structure for aperiodic sequences with this maximal pattern complexity.

We introduce Feldman–Katok convergence for invariant measures of a topological dynamical system. This can be seen as a counterpart to the convergence with respect to the $\bar {f}$-metric for finite-state stationary processes (shift-invariant measures on a symbolic space). Feldman–Katok convergence is based on a dynamically defined Feldman–Katok pseudometric. This convergence is stronger than weak$^*$ convergence. We prove that Feldman–Katok convergence preserves ergodicity and makes the Kolmogorov–Sinai entropy lower semicontinuous, thereby preserving zero entropy. We apply our findings to non-hyperbolic (having at least one vanishing Lyapunov exponent) ergodic measures constructed using the GIKN method as axiomatized by Bonatti, Díaz and Gorodetski [Nonlinearity, 23 (2010), 687–705]. The GIKN method, originally introduced by Gorodetski, Ilyashenko, Kleptsyn and Nalsky [Functional Analysis and its Applications, 39 (2005), 21–30], has been widely adapted to produce non-hyperbolic ergodic measures for diffeomorphisms of compact manifolds. We prove that an ergodic measure satisfying the conditions provided by the axiomatized GIKN method is the Feldman–Katok limit of a sequence of periodic measures, which implies that it is either a periodic measure or a loosely Kronecker measure (a measure Kakutani equivalent to an aperiodic ergodic rotation on a compact group) and has zero entropy. This classifies all these measures up to Kakutani equivalence and confirms that geometric constructions of non-hyperbolic measures via periodic approximations based on the axiomatized GIKN method presented in Bonatti et al. [op. cit.] systematically produce zero-entropy systems.

Let $ ([0,1]^d,T,\mu ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under suitable conditions on the underlying dynamical system, we obtain a strong dynamical Borel–Cantelli lemma for recurrence: for any sequence $ \{R_n\} $ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to $0$ as $n\to \infty $, where $ \mathbf {x}\in [0,1]^d $ and $ R_n+\mathbf {x} $ is the translation of $ R_n $. The result applies to the Gauss map, $\beta $-transformations, and expanding toral endomorphisms.

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study the density of the polynomial orbits in Toeplitz systems—we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.

We show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty $-step pro-nilfactor $X_\infty $ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no non-trivial $(k+1)$-tuples with arbitrarily long finite IP-independence sets, then it has only at most k ergodic measures and is an almost $k'$ to one extension of $X_\infty $ for some $k'\leqslant k$. In particular, for $k=1$, we prove that $(X,T)$ is uniquely ergodic (even regular with respect to $X_\infty $), which answers a conjecture stated by Dong et al [Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33(1) (2013), 118–143].

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X. We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H-orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G, and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H-orbit to a single compact set in X. This complements our results in the case of an arithmetic lattice.

In this article, we study the pressure at infinity of potentials defined over countable Markov shifts. We establish an upper semi-continuity result concerning the limiting behaviour of the pressure of invariant probability measures, where the escape of mass is controlled by the pressure at infinity. As a consequence, we establish criteria for the existence of equilibrium states and maximizing measures for uniformly continuous potentials. Additionally, we study the pressure at infinity of suspension flows defined over countable Markov shifts and prove an upper semi-continuity result for the pressure map.