Gibbs measures on Cayley trees and homoclinic orbits

Source abstract

We study the question of existence and multiplicity of homogeneous Markov Gibbs measures on Cayley trees with spin space $\mathbb{Z}$. We prove that, under a natural summability assumption on the interaction function, there are always at least two homogeneous Markov Gibbs measures satisfying two natural symmetry conditions. Moreover, any two homogeneous Markov Gibbs measures satisfying a suitable non-degeneracy assumption can be superimposed and produce infinitely many other homogeneous Markov Gibbs measures. When the interaction function $Q$ has bounded support in $\mathbb{Z}$, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a symplectomorphism of $\mathbb{R}^{2n}$, where $n$ is the smallest natural number such that $Q(i)=0$ for every $i>n$. For $n=1$, results from symbolic dynamics and horseshoes imply the existence of homogeneous Markov Gibbs measures whose single-site marginal is concentrated near an arbitrary finite subset of $N\mathbb{Z}$, where $N$ is a sufficiently large integer. In the case of interaction functions with infinite support, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a Lagrangian correspondence on the cotangent bundle of a Banach space.