Complexity as a homeomorphism invariant for tiling spaces

TL;DR

The paper demonstrates that the complexity and repetitivity functions of aperiodic, repetitive tilings with finite local complexity are invariants under homeomorphism of their tiling spaces, linking topological properties to combinatorial complexity.

Contribution

It establishes that complexity functions are topological invariants for tiling spaces and extends this to $ ext{Z}^d$-subshifts, providing new insights into tiling deformations.

Findings

Complexity functions are asymptotically equivalent under homeomorphism.

Repetitivity functions are also invariant under homeomorphism.

Polynomial complexity exponents are preserved by homeomorphism.

Abstract

It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of $d$-dimensional infinite words: if two $\mathbb{Z}^d$-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent. An analogue theorem is proved for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space. How this result relates to the theory of tilings deformations is outlined in the last part.