## Final Report Summary - APPMSFT (AUTOMORPHISMS AND PERIODIC POINTS FOR MULTIDIMENSIONAL SHIFTS OF FINITE TYPE)

Symbolic dynamics (from a certain abstract approach) is the branch of topological dynamics concerned with expansive group actions on zero-dimensional compacta, their automorphisms and endomorphisms.

Roughly speaking, an action of a group $G$ on a compact metric space $X$ is expansive if

any pair of different points in the system can be distinguished by applying measurements of limited precision along finite (possibly large) orbits segments.

The field has classical applications in smooth dynamics, coding theory, automata theory and the study of formal languages.

Of particular importance are subshifts of finite type.

Up to isomorphism, subshifts of finite type coincide with the class of subshifts which can be described by specifying finitely many local adjacency rules. Shifts of finite type are particularly attractive as simple models for systems that show up in our real world.

In this project we study symbolic dynamical systems, most notably multidimensional shifts of finite type. ,

we are specifically concerned with the automorphism groups of multidimensional shifts of finite type, the periodic points (finite orbits) of such subshifts, and how the automorphism group acts on them.

The goals of this project, in a broad sense, are to find structural algebraic and geometric results about automorphisms and automorphism groups of symbolic dynamical systems and to devise new techniques for construction of automorphisms.

In the 1980's Krieger associated to any Z-subshift of finite type an automorphism of

a countable ordered abelian group, called the dimension group. The dimension group together with this automorphism constitute the dimension module.

Among many other important applications (for example understanding shift equivalence), the dimension module gives a very useful representation of the group of automorphisms into the automorphisms of the dimension module. This representation has been studied by in particular by Boyle, Lind and Rudolph.

Another very useful representation for the automorphism group is the so-called ``gyration representation'' of Boyle and Krieger, which takes values in an abelian group and is based on the action on finite orbits.

One goal of this project is to find meaningful and useful extensions for the representations above into the setting of multidimensional shifts of finite type.

A specific related question is the following: Given an automorphism of a finite system, when does it extend to an automorphism of the entire system? In the one dimensional setting it is known through the works of Kim-Roush-Wagoner and others that the answer is very much related to the dimension representation

One of the central goals of this research is to understand what ``basic types'' of automrophisms are there. The shifts themselves are an obvious kind of automorphism. More generally, if the system is a direct product of two systems, applying a shift to one of the ``factors'' lead to an automorphism. This lead to considering the question of when a system admits a ``direct-factorization'' of this type, and it so in what ways. For full-shifts, J. Kari related this question to the existence of ``block-representation'' for automorphisms.

The project webpage :

https://www.math.bgu.ac.il/~mtom/APPMSFT/APPMSFT.html

Roughly speaking, an action of a group $G$ on a compact metric space $X$ is expansive if

any pair of different points in the system can be distinguished by applying measurements of limited precision along finite (possibly large) orbits segments.

The field has classical applications in smooth dynamics, coding theory, automata theory and the study of formal languages.

Of particular importance are subshifts of finite type.

Up to isomorphism, subshifts of finite type coincide with the class of subshifts which can be described by specifying finitely many local adjacency rules. Shifts of finite type are particularly attractive as simple models for systems that show up in our real world.

In this project we study symbolic dynamical systems, most notably multidimensional shifts of finite type. ,

we are specifically concerned with the automorphism groups of multidimensional shifts of finite type, the periodic points (finite orbits) of such subshifts, and how the automorphism group acts on them.

The goals of this project, in a broad sense, are to find structural algebraic and geometric results about automorphisms and automorphism groups of symbolic dynamical systems and to devise new techniques for construction of automorphisms.

In the 1980's Krieger associated to any Z-subshift of finite type an automorphism of

a countable ordered abelian group, called the dimension group. The dimension group together with this automorphism constitute the dimension module.

Among many other important applications (for example understanding shift equivalence), the dimension module gives a very useful representation of the group of automorphisms into the automorphisms of the dimension module. This representation has been studied by in particular by Boyle, Lind and Rudolph.

Another very useful representation for the automorphism group is the so-called ``gyration representation'' of Boyle and Krieger, which takes values in an abelian group and is based on the action on finite orbits.

One goal of this project is to find meaningful and useful extensions for the representations above into the setting of multidimensional shifts of finite type.

A specific related question is the following: Given an automorphism of a finite system, when does it extend to an automorphism of the entire system? In the one dimensional setting it is known through the works of Kim-Roush-Wagoner and others that the answer is very much related to the dimension representation

One of the central goals of this research is to understand what ``basic types'' of automrophisms are there. The shifts themselves are an obvious kind of automorphism. More generally, if the system is a direct product of two systems, applying a shift to one of the ``factors'' lead to an automorphism. This lead to considering the question of when a system admits a ``direct-factorization'' of this type, and it so in what ways. For full-shifts, J. Kari related this question to the existence of ``block-representation'' for automorphisms.

The project webpage :

https://www.math.bgu.ac.il/~mtom/APPMSFT/APPMSFT.html