Exact renormalisation for patch frequencies in inflation systems

2025/07/10 by Mazáč, Jan
#Dynamical Systems (math.DS) #FOS: Mathematics #Primary 37B52 #Secondary 52C23

paper · doi:10.48550/arxiv.2507.07753

Abstract

This note provides an explicit way of calculating the patch frequencies in geometric realisations of primitive substitutions using exact renormalisation relations. Further, we profit from these results to obtain the patch frequencies in the symbolic case as well as in other suspensions (under mild assumptions on the substitution). We illustrate this procedure on the Fibonacci example.

Citations

• A spectral cocycle for substitution systems and translation flows
• Tiling deformations, cohomology, and orbit equivalence of tiling spaces
• Invariant measures for non-primitive tiling substitutions
• Weighted model sets and their higher point-correlations
• Invariant Measures on Stationary Bratteli Diagrams
• Pattern equivariant functions, deformations and equivalence of tiling spaces
• Uniform Distribution in Model Sets

Related

• Finite parts of inflationary loops
• Consistency relations and degeneracies in (non)commutative patch inflation
• Resolving the Inflationary Power Spectrum
• Scale-Dependent Loop Corrections to the Inflationary Power Spectrum
• Asymptotic Safety within on-shell perturbation theory