Bounds for Equilibrium States on Amenable Group Subshifts
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915374743683072 |
|---|---|
| author | Hedges, C. Evans |
| author_facet | Hedges, C. Evans |
| contents | We prove a result on equilibrium measures for potentials with summable variation on arbitrary subshifts over a countable amenable group. For finite configurations $v$ and $w$, if $v$ is always replaceable by $w$, we obtain a bound on the measure of $v$ depending on the measure of $w$ and a cocycle induced by the potential. We then use this result to show that under this replaceability condition, we can obtain bounds on the Lebesgue-Radon-Nikodym derivative $d (μ_ϕ\circ ξ) / dμ_ϕ$ for certain holonomies $ξ$ that generate the homoclinic (Gibbs) relation. As corollaries, we obtain extensions of results by Meyerovitch and Garcia-Ramos and Pavlov to the countable amenable group subshift setting. Our methods rely on the exact tiling result for countable amenable groups by Downarowicz, Huczek, and Zhang and an adapted proof technique from Garcia-Ramos and Pavlov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for Equilibrium States on Amenable Group Subshifts Hedges, C. Evans Dynamical Systems Mathematical Physics We prove a result on equilibrium measures for potentials with summable variation on arbitrary subshifts over a countable amenable group. For finite configurations $v$ and $w$, if $v$ is always replaceable by $w$, we obtain a bound on the measure of $v$ depending on the measure of $w$ and a cocycle induced by the potential. We then use this result to show that under this replaceability condition, we can obtain bounds on the Lebesgue-Radon-Nikodym derivative $d (μ_ϕ\circ ξ) / dμ_ϕ$ for certain holonomies $ξ$ that generate the homoclinic (Gibbs) relation. As corollaries, we obtain extensions of results by Meyerovitch and Garcia-Ramos and Pavlov to the countable amenable group subshift setting. Our methods rely on the exact tiling result for countable amenable groups by Downarowicz, Huczek, and Zhang and an adapted proof technique from Garcia-Ramos and Pavlov. |
| title | Bounds for Equilibrium States on Amenable Group Subshifts |
| topic | Dynamical Systems Mathematical Physics |
| url | https://arxiv.org/abs/2401.13878 |