arXiv subjects

Elżbieta Krawczyk

Publications and source records attributed to Elżbieta Krawczyk.

5 recordsLinked to original sources

arXiv2026

Tameness, nullness, and amorphic complexity of automatic systems

In topological dynamics, tame and null systems arise naturally in the study of low-complexity aperiodic behaviour, yet providing concrete and easily testable conditions to establish their existence in a canonical class of systems is often nontrivial. We give a complete characterisation of tameness and nullness for minimal automatic systems generated by primitive constant length substitutions in terms of amorphic complexity -- a numerical invariant recently introduced to study zero entropy systems. We derive an easily computable closed formula for this invariant in this setting and show that, for infinite automatic systems, tameness and nullness are equivalent to its value being one.

math.DS↗

arXiv2026

Automaticity of uniformly recurrent substitutive sequences

We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question posed by Allouche, Dekking and Queffélec in the uniformly recurrent case. We show that the same criterion characterizes automaticity of minimal substitutive systems. Furthermore, we construct a minimal substitutive system whose maximal equicontinuous factor is the 2-adic odometer, and for which the corresponding factor map is everywhere uncountable-to-one. We conjecture that a minimal substitutive system is k-automatic if and only if it is an everywhere finite-to-one extension of a k-adic odometer.

math.NT↗

arXiv2024

Quasi-fixed points of substitutive systems

We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within an automatic system are not themselves automatic. We provide a complete and succinct classification of automatic sequences that lie in a given automatic system in terms of the quasi-fixed points of the substitution defining the system. Our result extends to factor maps between automatic systems and highlights arithmetic properties underpinning these systems. We conjecture that a similar statement holds for general nonconstant length substitutions.

math.CO↗

arXiv2021

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

math.CO↗

arXiv2020

Substitutive systems and a finitary version of Cobham's theorem

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.

math.CO↗