# Synchronization is full measure for all $\alpha$-deformations of an infinite class of continued fraction transformations

## TL;DR

This paper investigates the synchronization phenomena in a broad family of $oldsymbol{eta}$-deformed continued fraction maps linked to Fuchsian groups, revealing that synchronization fully characterizes the entropy behavior across all deformations.

## Contribution

It explicitly determines the synchronization sets for an infinite family of $oldsymbol{eta}$-deformed continued fraction transformations, extending beyond classical modular group cases.

## Key findings

- Synchronization sets are described via a single tree of words.
- Synchronization fully determines the entropy variation in these deformations.
- The sets are explicitly characterized for each $oldsymbol{eta}$-deformation.

## Abstract

We study an infinite family of one-parameter deformations, so-called $\alpha$-continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the entropy of the map indexed by $\alpha$ varies in a way directly related to whether or not the orbits of the endpoints of the map synchronize. For two cases of one-parameter deformations associated to the classical case of the modular group $\text{PSL}_2(\mathbb Z)$, the set of $\alpha$ for which synchronization occurs has been determined. Here, we explicitly determine the synchronization sets for each $\alpha$-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $\alpha$-expansions back to regular continued fraction expansions is not available to us.) A curiosity here is that all of our synchronization sets can be described in terms of a single tree of words. In a paper in preparation, we identify the natural extensions of our maps, as well as the entropy functions associated to each deformation.