Density of orbits of semigroups of endomorphisms acting on the Adeles

TL;DR

This paper studies the density of orbits in the Adeles under semigroup actions, proving that certain multiplicative semigroups generate dense orbits for points with irrational real parts.

Contribution

It establishes conditions under which orbits are dense in the Adeles, specifically when the semigroup contains multiplicatively independent elements including an integer.

Findings

Orbits are dense for points with irrational real coordinates under specified semigroup actions.

Density holds when the semigroup contains at least two multiplicatively independent elements, one being an integer.

The result extends understanding of dynamical systems on the Adeles with algebraic and number-theoretic implications.

Abstract

We investigate the question of whether or not the orbit of a point in A/Q, under the natural action of a subset S of Q, is dense in A/Q. We prove that if the set S is a multiplicative semigroup which contains at least two multiplicatively independent elements, one of which is an integer, then the orbit under S of any point with irrational real coordinate is dense.