Exact renormalisation for patch frequencies in inflation systems

TL;DR

This paper introduces an explicit method for calculating patch frequencies in inflation systems using exact renormalisation, applicable to both geometric and symbolic cases, demonstrated through the Fibonacci example.

Contribution

It provides a novel explicit approach for computing patch frequencies in inflation systems via exact renormalisation relations, extending to symbolic and suspension cases.

Findings

Explicit calculation method for patch frequencies in geometric realizations.

Application of the method to the Fibonacci substitution system.

Extension of results to symbolic and suspension systems under mild assumptions.

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.