eigenmorphic

Eigenvalues and more for morphic subshifts

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Morphic subshifts are generalization of substitution subshifts, where we allow taking the image of a substitution subshift by another substitution, considering the generated subshift.

This eigenmorphic package for SageMath contains:

• computation of exact additive eigenvalues of morphic subshifts
• decide recognizability of a substitution in the subshift of another substitution
• test if a morphic subshift has pure discrete spectrum using an improvement of the balanced pair algorithm
• compute substitutions from an IET given as a Rauzy loop or with lengths
• plot very general Rauzy fractals
• plot fixed points of Anosov maps from IET
• compute coboundaries

Installation

sage -pip install eigenmorphic

Usage

sage: from eigenmorphic import *

After this command, you can compute eigenvalues of morphic subshifts

sage: s = WordMorphism('a->ab,b->ac,c->a')
sage: morphic_eigenvalues(s)
Z*{1, b, b^2}
where b is root of x^3 - x^2 - x - 1

sage: t = WordMorphism('a->0,b->1,c->1')
sage: morphic_eigenvalues(s, t)
Z*{1, b, b^2}
where b is root of x^3 - x^2 - x - 1

# regular paperfolding
sage: t = WordMorphism('a->00,b->01,c->10,d->11')
sage: s = WordMorphism('a->ca,b->cb,c->da,d->db')
sage: t(s.fixed_points()[0])
word: 1101100111001001110110001100100111011001...
sage: morphic_eigenvalues(s,t)
1/8Z[1/2]

There are tools to compute coboundaries

sage: s = WordMorphism('a->c,b->de,c->bde,d->b,e->deab')
sage: coboundary_basis(s)
[ 0  1  0 -1  0]
[ 0  0  0  1 -1]

You can also test if the Z-action of a morphic subshift has pure discrete spectrum, using an improvement of the balanced pair algorithm

sage: s = WordMorphism("a->ab,b->ac,c->a")
sage: has_pure_discrete_spectrum(s)
True
sage: t = WordMorphism('a->ab,b->a,c->a')
sage: has_pure_discrete_spectrum(s, t, verb=1)
The condition ensuring that there is enough eigenvalues is satisfied.
execute balanced_pair_algorithm with w = a...
execute balanced_pair_algorithm with w = ab...
execute balanced_pair_algorithm with w = aba...
execute balanced_pair_algorithm with w = abac...
balanced pair algorithm terminated conclusively with w = a

True

There are also tools to find Rauzy loop in the graph of graphs, and plot fixed points of the corresponding Anosov

sage: b = AA(2*cos(pi/7))
sage: v = [4*b^2 - 2*b - 9, -7*b^2 + 6*b + 12, 5*b^2 - 4*b - 9, -b + 2, -3*b^2 + b + 8, b^2 - 3]
sage: per = "643215"
sage: rauzy_loop_substitution(per, v, gets2=1)
(WordMorphism: 1->1416, 2->14232416, 3->142332416, 4->142416, 5->156, 6->15616,
 WordMorphism: 1->12345664321, 2->23432, 3->323, 4->4321234, 5->56, 6->6432156)
sage: plot_surface_with_fixed_pts(per, v)

There are also tools to plot very general Rauzy fractals, from any finite word and projection

sage: u = s.periodic_points()[0][0]
sage: V = usual_projection(s.incidence_matrix())
sage: rauzy_fractal_plot(u[:100000], V)

You can also decide recognizability

sage: s = WordMorphism("a->ab,b->ac,c->a")
sage: t = WordMorphism('a->ab,b->a,c->a')
sage: is_recognizable(t, s)
True