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.

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 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.

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 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.

An important question in dynamical systems is the classification problem, that is, the ability to distinguish between two isomorphic systems. In this work, we study the topological factors between a family of multidimensional substitutive subshifts generated by morphisms with uniform support. We prove that it is decidable to check whether two minimal aperiodic substitutive subshifts are isomorphic. The strategy followed in this work consists of giving a complete description of the factor maps between these subshifts. Then, we deduce some interesting consequences on coalescence, automorphism groups, and the number of aperiodic symbolic factors of substitutive subshifts. We also prove other combinatorial results on these substitutions, such as the decidability of defining a subshift, the computability of the constant of recognizability, and the conjugacy between substitutions with different supports.

We introduce the concept of ‘irrational paths’ for a given subshift and useit to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott–Tomforde–Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

Using tools from computable analysis, we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural systems one can think of are effective in this sense, including some group rotations, affine actions on the torus and finitely presented algebraic actions. We show that for finitely generated and recursively presented groups, every effective dynamical system is the topological factor of a computable action on an effectively closed subset of the Cantor space. We then apply this result to extend the simulation results available in the literature beyond zero-dimensional spaces. In particular, we show that for a large class of groups, many of these natural actions are topological factors of subshifts of finite type.

This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids. Among the aspects studied are different types of cohomologies, the rates of mixing given by the Ruelle spectrum of the hyperbolic map acting on this space, and solutions of the cohomological equation in primitive substitution subshifts for Hölder functions. The overarching theme is that considerations of $\alpha $-Hölder regularity on Cantor sets go a long way.

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $\bar {d}$ metric ($\bar {d}$-approachable shift spaces). The class of $\bar {d}$-approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar {d}$-approachability, together with a closely connected notion of $\bar {d}$-shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys. 43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $\bar {d}$-approachability and $\bar {d}$-shadowing. Here, we study further properties and connections between $\bar {d}$-shadowing and $\bar {d}$-approachability. We prove that $\bar {d}$-shadowing implies $\bar {d}$-stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar {d}$-shadowing property the Hausdorff pseudodistance ${\bar d}^{\mathrm {H}}$ between shift spaces induced by $\bar {d}$ is the same as the Hausdorff distance between their simplices of invariant measures with respect to the Hausdorff distance induced by Ornstein’s metric $\bar {d}$ between measures. We prove that without $\bar {d}$-shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results, including minimal examples and proximal examples of shift spaces with the $\bar {d}$-shadowing property. The existence of such shift spaces was announced in the earlier paper mentioned above. It shows that $\bar {d}$-shadowing indeed generalizes the specification property.

Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism $f:M\rightarrow M$ on an open invariant subset $O\subset M$, which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers p and q in base $pq$ to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

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.

We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2) 199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math. 384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Two asymptotic configurations on a full $\mathbb {Z}^d$-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb {Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to $\mathbb {Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

Given a countable group G and a G-flow X, a probability measure $\mu $ on X is called characteristic if it is $\mathrm {Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal G-flow, for any group G, which does not admit a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection of infinite subgroups $\{\Delta _i: i\in I\}$, when is there a faithful G-flow for which every $\Delta _i$ acts minimally?