Proximality in Pisot Tiling Spaces

Details Proximality in Pisot Tiling Spaces Proximality in Pisot Tiling Spaces

2005-09-02

arxiv.org/abs/math/0509051v1

Mathematics

Dynamical Systems

52 pages, uses curves.sty

Scientific paper

A substitution $\vp$ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot $\vp$ that satisfies a no cycle condition and for which the translation flow on the tiling space $\T_\vp$ has pure discrete spectrum, we describe the collection $\T^P_\vp$ of pairs of proximal tilings in $\T_\vp$ in a natural way as a substitution tiling space. We show that if $\psi$ is another such substitution, then $\T_\vp $ and $\T_\psi$ are homeomorphic if and only if $\T^P_\vp$ and $\T^P_\psi$ are homeomorphic. We make use of this invariant to distinguish tiling spaces for which other known invariants are ineffective. In addition, we show that for strong Pisot substitutions, pure discrete spectrum of the flow on the associated tiling space is equivalent to proximality being a closed relation on the tiling space.

Affiliated with

Also associated with

No associations

Rating

Proximality in Pisot Tiling Spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Proximality in Pisot Tiling Spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Proximality in Pisot Tiling Spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-193398