Invariant measures for non-primitive tiling substitutions

TL;DR

This paper studies invariant measures in non-primitive self-affine tiling systems, revealing that while ergodic probability measures are supported on minimal components, there exist other infinite invariant measures, with a complete characterization under mild conditions.

Contribution

It provides a comprehensive analysis of invariant measures for non-primitive tiling substitutions, including the characterization of finite and infinite measures and recognizability of non-periodic tilings.

Findings

Ergodic probability measures are supported on minimal components.

Existence of infinite, locally finite invariant measures.

Unique invariant measure supported on trivial periodic tilings.

Abstract

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $\sigma$-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the "integer Sierpi\'nski gasket and carpet" tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $\sigma$-finite invariant measure, which is locally finite and unique up to scaling.