Invariant measures for non-primitive tiling substitutions

M. I. Cortez, B. Solomyak

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

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 σ-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 Sierpiński gasket and carpet" tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

Original languageEnglish
Pages (from-to)293-342
Number of pages50
JournalJournal d'Analyse Mathematique
Volume115
Issue number1
DOIs
StatePublished - Jun 2011
Externally publishedYes

Bibliographical note

Funding Information:
∗Partially supported by Fondecyt 1100318 †Partially supported by NSF grant DMS-0654408.

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