Functions of Substitution Tilings as a Jacobian

TL;DR

This paper demonstrates that functions derived from primitive, star-shaped substitution tilings, including Penrose tilings, can be represented as Jacobians of biLipschitz homeomorphisms of the plane, linking tiling theory with geometric analysis.

Contribution

It establishes a novel connection between substitution tilings and Jacobian determinants of biLipschitz maps, extending to Penrose tilings.

Findings

Functions from primitive, star-shaped substitution tilings are Jacobians of biLipschitz homeomorphisms.

The result applies specifically to Penrose tilings.

Provides a new geometric interpretation of substitution tiling functions.

Abstract

In this paper we show that the function defined by a primitive, star shaped substitu- tion tiling of the plane, can be realized as a Jacobian of a biLipschitz homeomorphism of R^2. In particular it holds for any Penrose tiling.