Related papers: A dynamical characterization of C sets
H.Furstenberg and E.Glasner proved that for an arbitrary $k\in\mathbb{N}$, any piecewise syndetic set of integers contains a $k$-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in…
We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach…
We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental…
Using the algebraic structure of the Stone-Cech compactification of the integers, Furstenberg and Glasner proved that for arbitrary k, every piecewise syndetic set contains a piecewise syndetic set of k-term arithmetic progressions. We…
It is shown how regular model sets can be characterized in terms of regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$…
A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper…
The most powerful formulation of the Central Sets Theorem in an arbitrary semigroup was proved in the work of De, Hindman, and Strauss. The sets which satisfy the conclusion of the above Central Sets Theorem are called $C$-sets. The…
We systematize and analyze some results obtained in Subset Combinatorics of $G$ groups after publications the previous surveys [1-4]. The main topics: the dynamical and descriptive characterizations of subsets of a group relatively their…
This paper is devoted to studying the localization of mixing property via Furstenberg families. It is shown that there exists some $\mathscr{F}_{pubd}$-mixing set in every dynamical system with positive entropy, and some…
We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-\v{C}ech…
Model sets are always Meyer sets, but not vice-versa. This article is about characterizing model sets (general and regular) amongst the Meyer sets in terms of two associated dynamical systems. These two dynamical systems describe two very…
Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are…
Let K be an algebraically closed field of prime characteristic p, let N be a positive integer, let f be a self-map on the algebraic torus T=G_m^N defined over K, let V be a curve in T defined over K, and let x be a K-point of T. We show…
Extending the work of Cuntz and Vershik, we develop a general notion of independence for commuting group endomorphisms. Based on this concept, we initiate the study of irreversible algebraic dynamical systems, which can be thought of as…
We treat here the interrelation between formal languages and those dynamical systems that can be described by cellular automata (CA). There is a well-known injective map which identifies any CA-invariant subshift with a central formal…
A mathematical concept is identified and analyzed that is implicit in the 2012 paper Turing Incomputable Computation, presented at the Alan Turing Centenary Conference (Turing-100, Manchester). The concept, called dynamic level sets, is…
A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
We characterize relative notions of syndetic and thick sets using, what we call, "derived" sets along ultrafilters. Manipulations of derived sets is a characteristic feature of algebra in the Stone-\v{C}ech compactification and its…
Furstenberg, Glasscock, Bergelson, Beiglboeck have been studied abundance in arithmatic progression on various large sets like piecewise syndetic, central, thick, etc. but also there are so many sets in which abundance in progression is…