Quantum arithmetic dynamics

TL;DR

This paper explores the dynamics of Lattès maps in the complex plane through Cuntz-Krieger algebras, reducing their iterations to subshifts of finite type and calculating their zeta functions.

Contribution

It introduces a novel approach linking Lattès map dynamics with non-commutative algebraic structures and subshifts, enabling explicit zeta function computation.

Findings

Reduced Lattès map iterations to subshifts of finite type

Calculated the zeta function for Lattès maps

Established a connection between complex dynamics and non-commutative algebra

Abstract

We study dynamics of the Latt\`es maps in the complex plane in terms of the Cuntz-Krieger algebras associated to the endomorphisms of the non-commutative tori. In particular, it is shown that iterations of the Latt\`es maps can be reduced to the dynamics of the subshifts of finite type. Using such a reduction, we calculate the zeta function of the Latt\`es maps.