Gibbs measures on Cayley trees and homoclinic orbits
Abstract
We study the question of existence and multiplicity of homogeneous Markov Gibbs measures on Cayley trees with spin space 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 ℤ, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a symplectomorphism of ℝ2n, 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 Nℤ, 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.
Related
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BibTeX
@misc{abbondandolo2026gibbs,
title = {{Gibbs measures on Cayley trees and homoclinic orbits}},
author = {Alberto Abbondandolo and Pietro Majer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33287},
eprint = {2610.33287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33287},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gibbs measures on Cayley trees and homoclinic orbits
AU - Alberto Abbondandolo
AU - Pietro Majer
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33287
UR - https://arxiv.org/abs/2610.33287
AB - We study the question of existence and multiplicity of homogeneous Markov Gibbs measures on Cayley trees with spin space 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 ℤ, there is a one-to-one correspondence between homogeneous Markov Gibbs measures and suitable homoclinic orbits of a symplectomorphism of 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 Nℤ, 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.
ER -