Translation Numbers Define Generators of Fk+→Homeo+(𝕊1)

Abstract

We consider a minimal action of a finitely generated semigroup by homeomorphisms of a circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov's theorem and its corollaries.

Random iterations of homeomorphisms on the circle

“…3 Rotation numbers are also important when studying an IFS (which can be considered as a special group action) of orientation-preserving circle homeomorphisms. The surveys [12,19] review these facts, see also [13].…”

“…Intuitively we may in all cases regard the infimum of all positive elements of L = L(F, d) as the "common prime period" of all maps, where the case when L is infinite corresponds to a degenerated case. As mentioned above, for orientationpreserving homeomorphisms this number can be compared with the rotation number functions in [12,19,13].…”

“…As an immediate corollary of Proposition 1 and Theorem 1 we get the following result. This type of result is usually referred to as Antonov's theorem (see [2], where all maps in the IFS are assumed to preserve orientation, see also [13,14]). Also in our generality, the present corollary is not new and follows (although not explicitly stated) from results by Malicet [17] who studied an even more general setting ( without assuming minimality).…”

“…In contrast to many previous authors we do not assume that all maps preserve orientation or, a priori, that the system of inverse maps is forward minimal (such as in [1,12,14,19,13,21]) or contains at least one map which is minimal (as in [21]). Our setting is also studied in [17] (without any minimality condition), where a different approach is used and ideas of [3] are adapted which in turn are built on ideas of [15,8].…”

“…This topic has been studied already from a number of different points of view. One may, for example, take a purely deterministic approach and study the associated action of the group of circle homeomorphisms (the special case of the group of orientation preserving circle diffeomorphisms is treated in [12,19,13]). Or one may, as we will, take a probabilistic approach and investigate Markov chains generated by random independent and identically distributed (i.i.d.)…”