Fixed points of Lyapunov integral operators and Gibbs measures

TL;DR

This paper explores the relationship between Lyapunov integral operators and Gibbs measures in models with uncountable spin values on Cayley trees, establishing fixed point existence and conditions for uniqueness.

Contribution

It introduces new results on fixed points of Lyapunov integral operators and their connection to Gibbs measures for models with uncountable spins.

Findings

Proved existence of fixed points for Lyapunov integral operators.

Established conditions for the uniqueness of fixed points.

Connected fixed points to Gibbs measures in uncountable spin models.

Abstract

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for four competing interactions of models with uncountable (i.e. $[0,1]$) set of spin values on a Cayley tree. And we shall prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of fixed points.