Nivat’s conjecture asserts that every two-dimensional configuration η:ℤ2→ whose rectangular pattern complexity satisfies Pη(k,n)≤kn for some k,n∈ℕ is periodic. A theorem of Cyr and Kra \cite{CyrKra16} establishes the conjecture in the short-rectangle case Pη(k,n)≤kn, with k≤3. Using the algebraic framework of Kari-Szabados \cite{KariSzabados20} and recent advances on periodic decompositions and one-sided nonexpansive directions \cite{Colle23,Colle22}, we extend the Cyr-Kra result to the case Pη(4,n)≤4n: every configuration satisfying this complexity bound is periodic. The key new ingredient is an intermediate structural theorem of independent interest: for any non-periodic configuration with low convex pattern complexity and integer-valued alphabet  contained in ℤ+, there exist a configuration ϑ in the orbit closure of η, a ℤ-minimal periodic decomposition ϑ=ϑ1+⋯+ϑm, a prime p∈ℕ with ⊂[[p]], and pairs of disjoint half-planes Ui,Vi⊂ℤ2 such that the reductions modulo p of the components ϑi are fully periodic on Ui and on Vi simultaneously, for each 1≤i≤m.

We introduce inner-distal homeomorphisms of compact metric spaces, namely homeomorphisms whose proximal cells have empty interior. We show that inner-distality imposes nontrivial dynamical restrictions on perfect compact metric spaces: every transitive inner-distal homeomorphism has a set of periodic points with empty interior. We place inner-distality within the hierarchy of generalized distality notions, proving that countably distal homeomorphisms are inner-distal and that every cw–distal homeomorphism on a perfect locally connected compact metric space is inner-distal. We also establish preservation under semi-open extensions and invariance under topological conjugacy. Motivated by Baire category, we introduce the stronger notion of meagre-distality. We prove that meagre-distal homeomorphisms admit uncountably many almost periodic points and stable classes, cannot be densely Li–Yorke chaotic, and do not satisfy the two-sided limit shadowing property.

Abstract: Given a language generated by an automaton over an alphabet of real (or complex) numbers, and a real or complex number  \beta in the open unit disk, we consider the \beta-direct valuation and reversed valuation sets of this language. For the k-Pell, k-Fibonacci and k-metallic languages, we compute and describe the topological properties of these valuation sets when  is the smaller root of the polynomial associated with these languages. Furthermore, we introduce the connectivity locus for the valuations of these languages, establish bounds for these sets, and exhibit some points in their boundary.

Abstract: For upper semi-continuous potentials defined on shifts over countable
alphabets, this paper ensures sufficient conditions for the existence of a maximizing
measure. We resort to the concept of blur shift, introduced by T. Almeida and
M. Sobottka as a compactification method for countable alphabet shifts consisting
of adding new symbols given by blurred subsets of the alphabet. Our approach
extends beyond the Markovian case to encompass more general countable alphabet
shifts. In particular, we guarantee a convex characterization and compactness for
the set of blur invariant probabilities with respect to the discontinuous shift map.
https://arxiv.org/abs/2507.18736

Abstract: 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 ℤd-shift on a non-trivial alphabet if and only if d≥2.
https://arxiv.org/abs/2303.17270

Abstract: We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. For these structures, which we call blueprints, we provide a general framework for symbolic dynamics under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that a variant of the domino problem for geometric tilings of ℝd is undecidable for d≥2 on any underlying tiling space with finite local complexity.
https://arxiv.org/abs/2504.05194

Abstract: In 2023, two striking, nearly simultaneous, mathematical discoveries have excited their respective communities, one by Greenfeld and Tao, the other (the Hat tile) by Smith, Myers, Kaplan and Goodman-Strauss, which can both be summed up as the following: there exists a single tile that tiles, but not periodically (sometimes dubbed the einstein problem). The two settings and the tools are quite different (as emphasized by their almost disjoint bibliographies): one in Euclidean geometry, the other in group theory. Both are highly nontrivial: in the first case, one allows complex shapes; in the second one, also the space to tile may be complex.
We propose here a framework that embeds both of these problems. From any tile system in this general framework, with some natural additional conditions, we exhibit a construction to simulate it by a group-theoretical tiling. We illustrate this by transforming the Hat tile into a new aperiodic group monotile, and we describe the symmetries of both the geometrical Hat tilings and the group tilings we obtain.
https://arxiv.org/abs/2409.15880