Abstract We consider non-autonomous conformal iterated function systems (NACIFSs) and their limit sets. Our main focus is on harmonic measure and its dimensions: Hausdorff and packing. We prove that these two dimensions are continuous under perturbations and that they satisfy Bowen’s and Manning’s type formulas. To achieve this, we establish general results about measures, and more broadly about positive functionals, defined on a symbolic space. We also develop tools from thermodynamic formalism in a non-autonomous setting.

Abstract 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.

Abstract We show that the canonical anticommutation relations (CAR) algebra admits a Cantor spectrum C ∗ $\mathrm {C}^\ast $ normal upper C Superscript asterisk -diagonal that is not conjugate to the standard AF diagonal. We obtain this by classification theory of C ∗ $\mathrm {C}^\ast $ normal upper C Superscript asterisk -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.

Abstract We classify the sets of natural numbers n for which certain dynamical systems ( X , f ) $(X,f)$ left parenthesis upper X comma f right parenthesis 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.

Abstract We show that the continuity property of Lyapunov exponents proved by Buzzi et al. [Continuity properties of Lyapunov exponents for surface diffeomorphisms. Invent. Math. 230 (2) (2022), 767–849] for smooth surface diffeomorphisms extends to smooth interval maps in the case when the map only has non-flat critical points and the entropies converging to the topological entropy. The result we obtained is stronger than the continuity of Lyapunov exponents, in particular, we prove the uniform integrability of Lyapunov exponents over entropies.

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Abstract We establish a relation between the continuity of the fiber entropy and the continuity of the fiber Lyapunov exponents for skew products with two-dimensional fibers. This result extends the theorem for surfaces proved by Buzzi, Crovisier, and Sarig. As a consequence, we are able to obtain classes of skew products that satisfy the strong positive recurrence property, in particular, these maps have finite number of measures of maximal entropy, all exponentially mixing with good statistical properties.

Abstract We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors [Chousionis, Leykekhman, and Urbański. The dimension spectrum of conformal graph directed Markov systems. Selecta Math. (N.S.) 25 (2019), 74] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed [Chousionis et al . Rigorous Hausdorff dimension estimates for conformal fractals. Preprint , 2024, arXiv:2408.06330]. Another key ingredient in our proof is obtaining, for the first time, reasonably sized distortion constants for the (infinite) Apollonian iterated function system.

Abstract 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.