Amalgamated Free Products of Circle Actions with a Bounded Number of Fixed Points

TL;DR

This paper introduces a new construction called amalgamated free product of circle actions, analyzing its dynamical properties and fixed point bounds, with implications for conjugacy and M"obius-like actions.

Contribution

It defines the amalgamated free product of circle actions, establishes conditions for minimality and fixed point bounds, and explores conjugacy and M"obius-like properties.

Findings

Construction yields a well-defined, minimal circle action.

Provides criteria for bounded fixed points, at most 2n.

Conditions for M"obius-like actions and non-conjugacy to finite lifts.

Abstract

Inspired by constructions of Kova\v{c}evi\'{c}, we introduce the amalgamated free product of circle actions, obtained by blowing up two actions along prescribed orbits and rearranging the inserted intervals. Under natural orbit and index assumptions, we prove that this construction is well defined, yields a minimal action on the circle, and is unique up to topological conjugacy. We then study its dynamical properties. Using a proper ping-pong partition arising from the construction, we obtain criteria ensuring that the resulting action still has a uniformly bounded number of fixed points, and in particular at most \(2n\) fixed points. We also give sufficient conditions for the resulting action to remain M\"obius-like and for it not to be topologically conjugate to a subgroup of any finite lift \(\psl^{(k)}(2,\RR)\).