# Teichm\"uller spaces of generalized symmetric homeomorphisms

## TL;DR

This paper introduces a new class of symmetric homeomorphisms on the unit circle, generalizing existing concepts, and establishes a complex Banach manifold structure for their space through barycentric extension and biholomorphic automorphisms.

## Contribution

It defines generalized symmetric homeomorphisms on the circle and constructs a complex Banach manifold structure for their space.

## Key findings

- The new class extends classical symmetric homeomorphisms.
- A complex Banach manifold structure is established.
- Barycentric extension and biholomorphic automorphisms are key tools.

## Abstract

We introduce the concept of a new kind of symmetric homeomorphisms on the unit circle, which is derived from the generalization of symmetric homeomorphisms on the real line. By the investigation of the barycentric extension for this class of circle homeomorphisms and the biholomorphic automorphisms induced by trivial Beltrami coefficients, we endow a complex Banach manifold structure on the space of those generalized symmetric homeomorphisms.