Conjugacy of aperiodic parts of shifts

AbbreviationThe full 2-shift#✓fully proved

\(X\) is the space of all two-sided sequences \(x = (x_i)_{i \in \mathbb{Z}}\) with every \(x_i \in \{0, 1\}\), carrying the product topology.

abbrev X : Type := FullShiftSpace 2

Sources

• [AMR-043-0001].

Uses 1 · Used by 3

Statement usesFull shift space

Used byAperiodic part of the full 2-shift · Hochman's aperiodic conjugacy question · Shift on the aperiodic part of the full 2-shift

AbbreviationThe full 3-shift#✓fully proved

The space of all two-sided sequences \(y = (y_i)_{i \in \mathbb{Z}}\) with every \(y_i \in \{0, 1, 2\}\), carrying the product topology. It is the ambient space of the proper 3-colouring shift.

abbrev YAmb : Type := FullShiftSpace 3

Sources

• [AMR-043-0001].

Uses 1 · Used by 4

Statement usesFull shift space

Used byAperiodic part of the 3-colouring shift · Hochman's aperiodic conjugacy question · Shift on the aperiodic part of the 3-colouring shift · The proper 3-colouring shift

DefinitionThe proper 3-colouring shift#✓fully proved

\(Y\) is the set of sequences \(y = (y_i)_{i \in \mathbb{Z}}\) in the full 3-shift (so every \(y_i \in \{0, 1, 2\}\)) such that \(y_i \ne y_{i+1}\) for every \(i \in \mathbb{Z}\).

It is a mixing shift of finite type.

def Y : Set YAmb := { y | ∀ i : ℤ, y i ≠ y (i + 1) }

Sources

• [AMR-043-0001].

Uses 1 · Used by 2

Statement usesThe full 3-shift

Used byAperiodic part of the 3-colouring shift · Shift on the aperiodic part of the 3-colouring shift

DefinitionAperiodic part of the full 2-shift#✓fully proved

Let \(\sigma\) be the left shift on the full 2-shift \(X\), given by \((\sigma x)_i = x_{i+1}\). The aperiodic part of \(X\) is the set of all \(x \in X\) that are not periodic for \(\sigma\), that is, with \(\sigma^n x \ne x\) for every \(n \ge 1\).

def AperiodicX : Set X := (Function.periodicPts (unitShift 2))ᶜ

Sources

• [AMR-043-0001].

Uses 2 · Used by 2

Statement usesThe full 2-shift · Unit shift

Used byHochman's aperiodic conjugacy question · Shift on the aperiodic part of the full 2-shift

DefinitionAperiodic part of the 3-colouring shift#✓fully proved

Let \(\sigma\) be the left shift on the full 3-shift, given by \((\sigma y)_i = y_{i+1}\). The aperiodic part of the proper 3-colouring shift \(Y\) is the set of all \(y \in Y\) that are not periodic for \(\sigma\), that is, with \(\sigma^n y \ne y\) for every \(n \ge 1\).

def AperiodicY : Set YAmb := Y \ Function.periodicPts (unitShift 3)

Sources

• [AMR-043-0001].

Uses 3 · Used by 2

Statement usesThe proper 3-colouring shift · The full 3-shift · Unit shift

Used byHochman's aperiodic conjugacy question · Shift on the aperiodic part of the 3-colouring shift

DefinitionShift on the aperiodic part of the full 2-shift#✓fully proved

The left shift \(\sigma\), given by \((\sigma x)_i = x_{i+1}\), maps the aperiodic part of the full 2-shift \(X\) into itself; this is the self-map of the aperiodic part of \(X\) obtained by restricting \(\sigma\) to it.

def shiftXRes (a : ↥AperiodicX) : ↥AperiodicX :=
  ⟨unitShift 2 a, by
    exact fun h ↦ a.property
      ((unitShift_mem_periodicPts_iff (N := 2) (a : X)).mp h)⟩

Sources

• [AMR-043-0001].

Uses 5 · Used by 1

Statement usesAperiodic part of the full 2-shift · The full 2-shift · Full shift space · The unit shift as a homeomorphism · Unit shift

Used byHochman's aperiodic conjugacy question

DefinitionShift on the aperiodic part of the 3-colouring shift#✓fully proved

The left shift \(\sigma\), given by \((\sigma y)_i = y_{i+1}\), maps the aperiodic part of the proper 3-colouring shift \(Y\) into itself; this is the self-map of the aperiodic part of \(Y\) obtained by restricting \(\sigma\) to it.

def shiftYRes (a : ↥AperiodicY) : ↥AperiodicY :=
  ⟨unitShift 3 a, by
    have ha : (a : YAmb) ∈ Y ∧ (a : YAmb) ∉ Function.periodicPts (unitShift 3) :=
      a.property
    constructor
    · change ∀ i : ℤ, unitShift 3 (a : YAmb) i ≠ unitShift 3 (a : YAmb) (i + 1)
      intro i
      simpa [unitShift_apply, add_assoc, add_comm, add_left_comm] using ha.1 (1 + i)
    · exact fun h ↦ ha.2
        ((unitShift_mem_periodicPts_iff (N := 3) (a : YAmb)).mp h)⟩

Sources

• [AMR-043-0001].

Uses 6 · Used by 1

Statement usesAperiodic part of the 3-colouring shift · The proper 3-colouring shift · The full 3-shift · Full shift space · The unit shift as a homeomorphism · Unit shift

Used byHochman's aperiodic conjugacy question

DefinitionHochman's aperiodic conjugacy question#✓fully proved

Let \(X\) be the full 2-shift and \(Y\) the proper 3-colouring shift, write \(X \setminus \mathrm{Per}(X)\) and \(Y \setminus \mathrm{Per}(Y)\) for their aperiodic parts with the subspace topologies, and let \(\sigma\) be the left shift \((\sigma z)_i = z_{i+1}\) on each. The statement is that there is a homeomorphism \(e \colon X \setminus \mathrm{Per}(X) \to Y \setminus \mathrm{Per}(Y)\) with \(e(\sigma a) = \sigma(e(a))\) for every \(a \in X \setminus \mathrm{Per}(X)\), that is, the two aperiodic parts are topologically conjugate.

def conjecture : Prop :=
  ∃ e : Homeomorph ↥AperiodicX ↥AperiodicY,
    ∀ a : ↥AperiodicX, e (shiftXRes a) = shiftYRes (e a)

Sources

• [AMR-043-0001] Hochman problem 1.

Uses 6 · Used by 1

Statement usesAperiodic part of the full 2-shift · Aperiodic part of the 3-colouring shift · The full 2-shift · The full 3-shift · Shift on the aperiodic part of the full 2-shift · Shift on the aperiodic part of the 3-colouring shift

Used byThe aperiodic parts of the full 2-shift and the 3-colouring shift are not conjugate

TheoremThe aperiodic parts of the full 2-shift and the 3-colouring shift are not conjugate#●ready to prove

Let \(X\) be the full 2-shift, \(Y\) the proper 3-colouring shift, and \(\sigma\) the left shift on each. There is no homeomorphism \(e\) from the aperiodic part of \(X\) onto the aperiodic part of \(Y\), each with the subspace topology, such that \(e(\sigma a) = \sigma(e(a))\) for every \(a\) in the aperiodic part of \(X\); that is, Hochman's aperiodic conjugacy question has a negative answer.

Salo proves that every conjugacy between infinite transitive shifts of finite type minus periodic points extends to a conjugacy of the shifts of finite type; since the full 2-shift has fixed points and the proper 3-colouring shift has none, the aperiodic parts are not conjugate.

public theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• Ville Salo, Conjugacy of transitive SFTs minus periodic points, Proc. London Math. Soc. (accepted; arXiv v3 2023), arXiv:2104.09860, https://arxiv.org/abs/2104.09860.

Uses 1

Statement usesHochman's aperiodic conjugacy question

8 fully proved1 ready to prove

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