Local Rules for Planar Computable Tilings

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Thomas Fernique & Mathieu Sablik

Bastia, September 20th, 2012

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The Problem The Tool The Proof

1 The Problem

2 The Tool

3 The Proof

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1 The Problem

2 The Tool

3 The Proof

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Canonical n → d tilings

n pairwise non-colinear vectors ofR^d _dⁿ

tiles tiling ofR^d.

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Canonical n → d tilings

Lift: homeomorphism which maps tiles on d-faces of unit n-cubes.

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Planarity

Planar: lift inE+ [0,t]ⁿ, whereE is the slope andt the thickness.

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Planarity

Perfect: planar with the minimal thicknesst= 1.

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Local rules

Definition

A slopeE admits local rulesif there is a finite set of patterns s.t.

any canonical tiling without these patterns is planar with slopeE.

Local rules are said to be

strongif the tilings satisfying them are perfect;

natural if the perfect tilings satisfy them;

weak otherwise (the thickness is thus just bounded).

Local rules can also bedecorated, with a tile playing different roles.

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The Computability barrier

Computable number: within precisionεby a Turing machine.

Proposition

If a slope admits local rules (of any type), then it is computable.

Proof sketch:

lett be the thickness (assumed to be known if rules are weak);

let εbe the wanted precision;

form a pattern covering a ball of radius r ≥t/ε;

taked free vectors of length r in this pattern;

they span a space at distance less than t/r from the slope.

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1 The Problem

2 The Tool

3 The Proof

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Subshifts

Definition

A nD subshift over an alphabetAis a translation invariant closed subset ofA^Zⁿ, whered(u,v) = 2^−sup^k≥0^u^|[−k,k]ⁿ^=v^|[−k,k]ⁿ.

A subshift is said to be

effective if its forbidden patterns are recursively enumerables;

of finite typeif defined by finitely manyforbidden patterns;

sofic if it is a letter-to-letter image of a subshift of finite type.

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Projective subaction

Theorem (Aubrun-Sablik’10, Durand-Romashchenko-Shen’10) If X⊂ A^Z is effective, then{y∈ A^Z², ∀j, y_j =y0∈X} is sofic.

Proof sketch:

Take the Robinson tiles;

Add a layer which allows to vertically repeat any line in A^Z; Enumerate the forbidden patterns in the Robinson boards;

Check that no forbidden pattern appears on the repeated line.

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1 The Problem

2 The Tool

3 The Proof

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Quasi-Sturmian words

Sturmian words_ρ,α ∈ {0,1}^Z of slopeα∈[0,1] and interceptρ:

sρ,α(n) = 0 ⇔ (ρ+nα) mod 1∈[0,1−α).

Distance over words in{0,1}^Z: d(u,v) := sup

p≤q

||u(p)· · ·u(q)|₀− |v(p)· · ·v(q)|₀|.

Proposition (Morse-Hedlund, 1940)

Two Sturmian words with the same slope are at distance at most1.

Definition

Quasi-Sturmian word: at distance at most 1 from a Sturmian word.

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Quasi-Sturmian words

Sturmian words_ρ,α ∈ {0,1}^Z of slopeα∈[0,1] and interceptρ:

sρ,α(n) = 0 ⇔ (ρ+nα) mod 1∈[0,1−α).

Distance over words in{0,1}^Z: d(u,v) := sup

p≤q

||u(p)· · ·u(q)|₀− |v(p)· · ·v(q)|₀|.

Proposition (Morse-Hedlund, 1940)

Two Sturmian words with the same slope are at distance at most1.

Definition

Quasi-Sturmian word: at distance at most 1 from a Sturmian word.

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Stripes of perfect 3 → 2 tilings

Perfect 3→2 tiling: intertwined stripes encoding Sturmian words.

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Stripes of perfect 3 → 2 tilings

Parallel stripes encode quasi-Sturmian words with the same slope.

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A sofic subshift

Proposition

The Sturmian words of comput. slopeα form an effective subshift.

Proof sketch: compute patterns up to haven+ 1 factors of size n.

The following 2D subshift is thus sofic:

Zα ={y ∈ {0,1}^Z², ∀j, yj =y0 =sα,0}, and thequasi-Sturmian subshiftalso (use a “carry” layer):

Z_α⁰ ={y ∈ {0,1}^Z², ∀j, d(y_j,s_α,0)≤1}.

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From perfect 3 → 2 tilings to quasi-Sturmian subshifts

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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From quasi-Sturmian subshifts to planar 3 → 2 tilings

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Higher (co)dimensions

Perfectn →d tilings: (_dⁿ)

2

There is yet no complete characterization of these slopes.