In this work, we introduce the type and typeset invariants for equicontinuous group actions on Cantor sets; that is, for generalized odometers. These invariants are collections of equivalence classes of asymptotic Steinitz numbers associated to the action. We show the type is an invariant of the return equivalence class of the action. We introduce the notion of commensurable typesets and show that two actions which are return equivalent have commensurable typesets. Examples are given to illustrate the properties of the type and typeset invariants. The type and typeset invariants are used to define homeomorphism invariants for solenoidal manifolds.

For a connected Lie group G and an automorphism T of G, we consider the action of T on Sub$_G$, the compact space of closed subgroups of G endowed with the Chabauty topology. We study the action of T on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of G. We relate the distality of the T-action on Sub$^p_G$ with that of the T-action on G and characterise the same in terms of compactness of the closed subgroup generated by T in Aut$(G)$ when T acts distally on the maximal central torus and G is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$ and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group G acts distally on Sub$^p_G$ by conjugation if and only if G is either compact or is isomorphic to a direct product of a compact group and a vector group. Some of our results generalise those of Shah and Yadav.

Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst. 27 (2010), 383–388] and Shinoda [Nonlinearity 31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.

We establish some interactions between uniformly recurrent subgroups (URSs) of a group G and cosets topologies $\tau _{\mathcal {N}}$ on G associated to a family $\mathcal {N}$ of normal subgroups of G. We show that when $\mathcal {N}$ consists of finite index subgroups of G, there is a natural closure operation $\mathcal {H} \mapsto \mathrm {cl}_{\mathcal {N}}(\mathcal {H})$ that associates to a URS $\mathcal {H}$ another URS $\mathrm {cl}_{\mathcal {N}}(\mathcal {H})$, called the $\tau _{\mathcal {N}}$-closure of $\mathcal {H}$. We give a characterization of the URSs $\mathcal {H}$ that are $\tau _{\mathcal {N}}$-closed in terms of stabilizer URSs. This has consequences on arbitrary URSs when G belongs to the class of groups for which every faithful minimal profinite action is topologically free. We also consider the largest amenable URS $\mathcal {A}_G$ and prove that for certain coset topologies on G, almost all subgroups $H \in \mathcal {A}_G$ have the same closure. For groups in which amenability is detected by a set of laws (a property that is variant of the Tits alternative), we deduce a criterion for $\mathcal {A}_G$ to be a singleton based on residual properties of G.

We study density properties of orbits for a hypercyclic operator T on a separable Banach space X, and show that exactly one of the following four cases holds: (1) every vector in X is asymptotic to zero with density one; (2) generic vectors in X are distributionally irregular of type $1$; (3) generic vectors in X are distributionally irregular of type $2\frac {1}{2}$ and no hypercyclic vector is distributionally irregular of type $1$; (4) every hypercyclic vector in X is divergent to infinity with density one. We also present some examples concerned with weighted backward shifts on $\ell ^p$ to show that all the above four cases can occur. Furthermore, we show that similar results hold for $C_0$-semigroups.

A tame dynamical system can be characterized by the cardinality of its enveloping (or Ellis) semigroup. Indeed, this cardinality is that of the power set of the continuum $2^{\mathfrak c}$ if the system is non-tame. The semigroup admits a minimal bilateral ideal and this ideal is a union of isomorphic copies of a group $\mathcal H$, called the structure group. For almost automorphic systems, the cardinality of $\mathcal H$ is at most ${\mathfrak c}$ that of the continuum. We show a partial converse of this which holds for minimal systems for which the Ellis semigroup of their maximal equicontinuous factor acts freely, namely that the cardinality of $\mathcal H$ is $2^{{\mathfrak c}}$ if the proximal relation is not transitive and the subgroup generated by products $\xi \zeta ^{-1}$ of singular points $\xi ,\zeta $ in the maximal equicontinuous factor is not open. This refines the above statement about non-tame Ellis semigroups, as it locates a particular algebraic component of the latter which has such a large cardinality.

In this article, we revisit the notion of some hyperbolicity introduced by Pujals and Sambarino [A sufficient condition for robustly minimal foliations. Ergod. Th. & Dynam. Sys. 26(1) (2006), 281–289]. We present a more general definition that, in particular, can be applied to the symplectic context (something that was not possible for the previous one). As an application, we construct $C^1$ robustly transitive derived from Anosov diffeomorphisms with mixed behaviour on centre leaves.

R. Pavlov and S. Schmieding [On the structure of generic subshifts. Nonlinearity 36 (2023), 4904–4953] recently provided some results about generic $\mathbb {Z}$-shifts, which rely mainly on an original theorem stating that isolated points form a residual set in the space of $\mathbb {Z}$-shifts such that all other residual sets must contain it. As a direction for further research, they pointed towards genericity in the space of $\mathbb {G}$-shifts, where $\mathbb {G}$ is a finitely generated group. In the present text, we approach this for the case of $\mathbb {Z}^d$-shifts, where $d \ge 2$. As it is usual, multidimensional dynamical systems are much more difficult to understand. In light of the result of R. Pavlov and S. Schmieding, it is natural to begin with a better understanding of isolated points. We prove here a characterization of such points in the space of $\mathbb {Z}^d$-shifts, in terms of the natural notion of maximal subsystems that we also introduce in this article. From this characterization, we recover the result of R. Pavlov and S. Schmieding for $\mathbb {Z}^1$-shifts. We also prove a series of results that exploit this notion. In particular, some transitivity-like properties can be related to the number of maximal subsystems. Furthermore, we show that the Cantor–Bendixon rank of the space of $\mathbb {Z}^d$-shifts is infinite for $d>1$, while for $d=1$, it is known to be equal to one.

We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts $B_w,B_{w'}$ on $c_0(\mathbb {Z}_+)$ whose direct sum ${B_w\oplus B_{w'}}$ is not $\mathcal {U}$-frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on $\ell _p(\mathbb {Z}_+)$, $1\le p<\infty $, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every $N\in \mathbb {N}$, any disjoint frequently hypercyclic N-tuple of operators $(T_1,\ldots ,T_N)$ can be extended to a disjoint frequently hypercyclic $(N+1)$-tuple $(T_1,\ldots ,T_N, T_{N+1})$ as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair $(D,\tau _a)$ is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and $\tau _a$ is the operator of translation by $a\in \mathbb {C}\setminus \{ 0\}$. Part of our results are in fact obtained in the general setting of Furstenberg families.

We study the descriptive complexity of sets of points defined by restricting the statistical behaviour of their orbits in dynamical systems on Polish spaces. Particular examples of such sets are the sets of generic points of invariant Borel probability measures, but we also consider much more general sets (for example, $\alpha $-Birkhoff regular sets and the irregular set appearing in the multifractal analysis of ergodic averages of a continuous real-valued function). We show that many of these sets are Borel in general, and all these are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is, in some cases, necessary to obtain Borelness. When these sets are Borel, we measure their descriptive complexity using the Borel hierarchy. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that these sets are properly located at the third level of the hierarchy for many dynamical systems. To demonstrate that the specification property is a sufficient, but not necessary, condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is $\boldsymbol {\Pi }^0_3$-complete.

We study the computational problem of rigorously describing the asymptotic behavior of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and we prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constraints and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non-computable.

Many continua that admit a transitive homeomorphism may be found in the literature. The circle is probably the simplest non-degenerate continuum that admits such a homeomorphism. However, most of the known examples of such continua have a complicated topological structure. For example, they are indecomposable (such as the pseudo-arc or the Knaster bucket-handle continuum), or they are not indecomposable but have some other complicated topological structure, such as a dense set of ramification points (such as the Sierpiński carpet) or a dense set of end-points (such as the Lelek fan). In this paper, we continue our mission of finding continua with simpler topological structures that admit a transitive homeomorphism. We construct a transitive homeomorphism on the Cantor fan. In our approach, we use two different techniques, each of them giving two constructions of a transitive homeomorphism on the Cantor fan: one technique using quotient spaces of products of compact metric spaces and Cantor sets, and one using Mahavier products of closed relations on compact metric spaces. We also demonstrate how our technique using Mahavier products of closed relations may be used to construct a transitive function f on a Cantor fan X such that $\varprojlim (X,f)$ is a Lelek fan.

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal {S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani–Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue–Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and, as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems.

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where I is a finite index set. Fix a nonempty convex subset $\mathscr {L}$ of W. If W is of type A, then $\mathscr {L}$ is the set of linear extensions of a poset, and there are important Bender–Knuth involutions $\mathrm {BK}_i\colon \mathscr {L}\to \mathscr {L}$ indexed by elements of I. For arbitrary W and for each $i\in I$, we introduce an operator $\tau _i\colon W\to W$ (depending on $\mathscr {L}$) that we call a noninvertible Bender–Knuth toggle; this operator restricts to an involution on $\mathscr {L}$ that coincides with $\mathrm {BK}_i$ in type A. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm {Pro}_c=\tau _{i_n}\cdots \tau _{i_1}$. We say W is futuristic if for every nonempty finite convex set $\mathscr {L}$, every Coxeter element c and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm {Pro}_c^K(u)\in \mathscr {L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When W is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of W, then $\tau _{i_N}\cdots \tau _{i_1}(W)=\mathscr {L}$; this allows us to determine the smallest integer $\mathrm {M}(c)$ such that $\mathrm {Pro}_c^{{\mathrm {M}}(c)}(W)=\mathscr {L}$ for all $\mathscr {L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.

We study scaled topological entropy, scaled measure entropy, and scaled local entropy in the context of amenable group actions. In particular, a variational principle is established.

Sets on the boundary of a complementary component of a continuum in the plane have been of interest since the early 1920s. Curry and Mayer defined the buried points of a plane continuum to be the points in the continuum which were not on the boundary of any complementary component. Motivated by their investigations of Julia sets, they asked what happens if the set of buried points of a plane continuum is totally disconnected and nonempty. Curry, Mayer, and Tymchatyn showed that in that case the continuum is Suslinian, i.e., it does not contain an uncountable collection of nondegenerate pairwise disjoint subcontinua. In an answer to a question of Curry et al., van Mill and Tuncali constructed a plane continuum whose buried point set was totally disconnected, nonempty, and one-dimensional at each point of a countably infinite set. In this paper, we show that the van Mill–Tuncali example was the best possible in the sense that whenever the buried set is totally disconnected, it is one-dimensional at each of at most countably many points. As a corollary, we find that the buried set cannot be almost zero-dimensional unless it is zero-dimensional. We also construct locally connected van Mill–Tuncali type examples.

We prove a result on equilibrium measures for potentials with summable variation on arbitrary subshifts over a countable amenable group. For finite configurations v and w, if v is always replaceable by w, we obtain a bound on the measure of v depending on the measure of w and a cocycle induced by the potential. We then use this result to show that under this replaceability condition, we can obtain bounds on the Lebesgue–Radon–Nikodym derivative $d (\mu _\phi \circ \xi ) / d\mu _\phi $ for certain holonomies $\xi $ that generate the homoclinic (Gibbs) relation. As corollaries, we obtain extensions of results by Meyerovitch [Gibbs and equilibrium measures for some families of subshifts. Ergod. Th. & Dynam. Sys. 33(3) (2013), 934–953], and García-Ramos and Pavlov [Extender sets and measures of maximal entropy for subshifts. J. Lond. Math. Soc. (2) 100(3) (2019), 1013–1033] to the countable amenable group subshift setting. Our methods rely on the exact tiling result for countable amenable groups by Downarowicz, Huczek, and Zhang [Tilings of amenable groups. J. Reine Angew. Math. 2019(747) (2019), 277–298] and an adapted proof technique from García-Ramos and Pavlov.

Krieger’s embedding theorem provides necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing $\mathbb {Z}$-subshift of finite type. For certain families of $\mathbb {Z}^d$-subshifts of finite type, Lightwood characterized the aperiodic subsystems. In the current paper, we prove a new embedding theorem for a class of subshifts of finite type over any countable abelian group. Our theorem provides necessary and sufficient conditions for an arbitrary subshift X to embed inside a given subshift of finite type Y that satisfies a certain natural condition. For the particular case of $\mathbb {Z}$-subshifts, our new theorem coincides with Krieger’s theorem. Our result gives the first complete characterization of the subsystems of the multidimensional full shift $Y= \{0,1\}^{\mathbb {Z}^d}$. The natural condition on the target subshift Y, introduced explicitly for the first time in the current paper, is called the map extension property. It was introduced implicitly by Mike Boyle in the early 1980s for $\mathbb {Z}$-subshifts and is closely related to the notion of an absolute retract, introduced by Borsuk in the 1930s. A $\mathbb {Z}$-subshift has the map extension property if and only if it is a topologically mixing subshift of finite type. We show that various natural examples of $\mathbb {Z}^d$ subshifts of finite type satisfy the map extension property, and hence our embedding theorem applies for them. These include any subshift of finite type with a safe symbol and the k-colorings of $\mathbb {Z}^d$ with $k \ge 2d+1$. We also establish a new theorem regarding lower entropy factors of multidimensional subshifts that extends Boyle’s lower entropy factor theorem from the one-dimensional case.

Let $(X,\mathcal {B},\mu ,T)$ be a probability-preserving system with X compact and T a homeomorphism. We show that if every point in $X\times X$ is two-sided recurrent, then $h_{\mu }(T)=0$, resolving a problem of Benjamin Weiss, and that if $h_{\mu }(T)=\infty $, then every full-measure set in X contains mean-asymptotic pairs (that is, the associated process is not tight), resolving a problem of Ornstein and Weiss.