← Back

Algebraic Numbers, Free Group Automorphisms and Substitutions on the Plane

An extension of the Pisot substitution framework to the non-Pisot hyperbolic case. Working through a specific automorphism of the free group on four generators.

2011-01-01

Authors
P. Arnoux, E. O. Harriss, S. Ito, M. Furukado
Published in
Transactions of the American Mathematical Society, 2011
Links
DOIPDF
BibTeX 
@article{algebraic-numbers-free-group,
  title   = {Algebraic Numbers, Free Group Automorphisms and Substitutions on the Plane},
  author  = {P. Arnoux and E. O. Harriss and S. Ito and M. Furukado},
  journal = {Transactions of the American Mathematical Society},
  year    = {2011},
  volume  = {363},
  pages   = {4651--4699},
  doi     = {10.1090/S0002-9947-2011-05188-3},
}

This paper justifies a sense that the Pisot framework is critical rather than the specifics of a Pisot number. In other words interesting things happen for hyperbolic linear maps when we consider the expanding space.

We construct a two 2d substitution rule from a simple automorphism that has an expanding and a contracting pair of complex eigenvalues.

σ:F4→F4,{1↦22↦33↦44↦4⋅1−1\sigma : F_4 \to F_4, \qquad \begin{cases} 1 \mapsto 2 \\ 2 \mapsto 3 \\ 3 \mapsto 4 \\ 4 \mapsto 4 \cdot 1^{-1} \end{cases}

The abelianization matrix is:

A=(000−1100001000011)A = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}

with characteristic polynomial PA(X)=X4−X3+1P_A(X) = X^4 - X^3 + 1.

This creates two stepped surfaces (for the automorphism and its inverse). Both tilings are shown to be projection tilings and the Markov partition for the hyperbolic action is constructed.