Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms

TL;DR

This paper investigates the conditions under which hyperbolic automorphisms of the 2-torus can be bijectively coded using arithmetic mappings, linking these conditions to binary quadratic forms and quadratic fields.

Contribution

It provides a necessary and sufficient condition for the existence of bijective arithmetic codings based on binary quadratic forms and characterizes all such codings via the Dirichlet group of quadratic fields.

Findings

Derived a condition for bijective coding existence using quadratic forms

Connected arithmetic codings to the structure of quadratic fields

Identified the minimal kernel order of the homomorphism

Abstract

We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto the 2-torus. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above.