We study an abstract group of reversible Turing machines. In our model, each machine is interpreted as a homeomorphism over a space which represents a tape filled with symbols and a head carrying a state. These homeomorphisms can only modify the tape at a bounded distance around the head, change the state, and move the head in a bounded way. We study three natural subgroups arising in this model: the group of finite-state automata, which generalizes the topological full groups studied in topological dynamics and the theory of orbit-equivalence; the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates, and the group of elementary Turing machines, which are the machines which are obtained by composing finite-state automata and oblivious Turing machines.

We show that both the group of oblivious Turing machines and that of elementary Turing machines are finitely generated, while the group of finite-state automata and the group of reversible Turing machines are not. We show that the group of elementary Turing machines has undecidable torsion problem. From this, we also obtain that the group of cellular automata (more generally, the automorphism group of any uncountable one-dimensional sofic subshift) contains a finitely generated subgroup with undecidable torsion problem. We also show that the torsion problem is undecidable for the topological full group of a full $\mathbb {Z}^d$-shift on a nontrivial alphabet if and only if $d \geq 2$.

We prove that Brinkmann’s problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\Phi \in \mathrm {End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb {N}$ such that $(x,y)\Phi ^k=(z,w)$ (or $(x,y)\Phi ^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.

We define a new class of plane billiards – the “pensive billiard” – in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so-called “puck billiards” proposed by M. Bialy, as well as a “vortex billiard,” that is, the motion of a point vortex dipole in two-dimensional hydrodynamics on domains with boundary. We prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards.

We define a family of discontinuous maps on the circle, called Bowen–Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor–Thurston theory for one-dimensional maps.

For full shifts on finite alphabets, Coelho and Quas [Criteria for $\overline {d}$-continuity. Trans. Amer. Math. Soc. 350(8) (1998), 3257–3268] showed that the map that sends a Hölder continuous potential $\phi $ to its equilibrium state $\mu _\phi $ is $\overline {d}$-continuous. We extend this result to the setting of full shifts on countable (infinite) alphabets. As part of the proof, we show that the map that sends a strongly positive recurrent potential to its normalization is continuous for potentials on mixing countable state Markov shifts.

This paper develops a geometric and analytical framework for studying the existence and stability of pinned pulse solutions in a class of non-autonomous reaction–diffusion equations. The analysis relies on geometric singular perturbation theory, matched asymptotic method and nonlocal eigenvalue problem method. First, we derive the general criteria on the existence and spectral (in)stability of pinned pulses in slowly varying heterogeneous media. Then, as a specific example, we apply our theory to a heterogeneous Gierer–Meinhardt (GM) equation, where the nonlinearity varies slowly in space. We identify the conditions on parameters under which the pulse solutions are spectrally stable or unstable. It is found that when the heterogeneity vanishes, the results for the heterogeneous GM system reduce directly to the known results on the homogeneous GM system. This demonstrates the validity of our approach and highlights how the spatial heterogeneity gives rise to richer pulse dynamics compared to the homogeneous case.

For continuous self-maps of compact metric spaces, we explore the relationship among the shadowable points, sensitive points, and entropy points. Specifically, we show that (1) if the set of shadowable points is dense in the phase space, then any interior point of the set of sensitive points is an entropy point; and (2) if the topological entropy is zero, then the denseness of the set of shadowable points is equivalent to almost chain continuity. In addition, we present a counter-example to a question raised by Ye and Zhang regarding entropy points.

In this paper, we establish variational principles for the metric mean dimension of random dynamical systems with infinite topological entropy. This is based on four types of measure-theoretic ϵ-entropies: Kolmogorov-Sinai ϵ-entropy, Shapira’s ϵ-entropy, Katok’s ϵ-entropy and Brin–Katok local ϵ-entropy. The variational principle, as a fundamental theorem, links topological dynamics and ergodic theory.

We introduce a new conjecture of global asymptotic stability for nonautonomous systems, which is fashioned along the nonuniform exponential dichotomy spectrum and whose restriction to the autonomous case is related to the classical Markus–Yamabe Conjecture: we prove that the conjecture is fulfilled for a family of triangular systems of nonautonomous differential equations satisfying boundedness assumptions. An essential tool to carry out the proof is a necessary and sufficient condition ensuring the property of nonuniform exponential dichotomy for upper block triangular linear differential systems. We also obtain some byproducts having interest on itself, such as the diagonal significance property in terms of the above-mentioned spectrum.

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.