# A Phase Transition for Circle Maps with a Flat Spot and Different Critical Exponents

## TL;DR

This paper investigates phase transitions in circle maps with flat spots and differing critical exponents by analyzing the asymptotic behavior of the renormalization operator, revealing how system geometry changes at boundary conditions.

## Contribution

It introduces a novel approach using renormalization operator asymptotics to study phase transitions in circle maps with flat intervals and varying critical exponents.

## Key findings

- Identification of a phase transition boundary depending on critical exponents
- Partition of system space into two connected regions separated by the boundary
- Asymptotic analysis of the renormalization operator elucidates geometric changes

## Abstract

We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.