Semigroups of real functions with dense orbits

TL;DR

This paper characterizes pairs of Möbius transformations that generate semigroups with dense orbits in an interval, providing a comprehensive description of when such dense orbits occur for all starting points.

Contribution

It offers a complete classification of pairs of functions in ${ m PSL}(2,R)$ that produce dense orbits in an interval for all initial points.

Findings

Identifies conditions under which orbits are dense for all points

Provides a characterization of semigroup generators with dense orbits

Enhances understanding of dynamical behavior of Möbius transformations

Abstract

Let ${\mathcal F}_I=\{f:I \to I| f(x)= (Ax+B)/(Cx+D); AD-BC \neq 0 \}$, where $I$ is an interval. For $x\in I$, let ${\Omega}_x$ be the orbit of $x$ under the action of the semigroup of functions generated by $f,g \in {\mathcal F}_I$. Our main result in this paper is to describe all $f,g \in {\mathcal F}_I$ such that $\Omega_x$ is dense in $I$ for all $x$.