Gottschalk's surjunctivity conjecture

A group $G$ is surjunctive if every injective, continuous, $G$-equivariant map $A^G \to A^G$ (for any finite alphabet $A$) is surjective.

Here equivariance is with respect to the left shift action of $G$ on $A^G$, defined by $(g \cdot x)(h) = x(g^{-1} h)$, and continuity is with respect to the product topology on $A^G$ (where $A$ carries the discrete topology).

Gottschalk's conjecture (1973) states that every group is surjunctive.

References:

• Wikipedia
• Gottschalk, W. H. (1973), "Some general dynamical notions"

def GottschalkSurjunctivity.shift (G : Type u_1) [Group G] {A : Type u_2} (g : G) (x : G → A) : G → A

The left shift of x : G → A by g : G, defined by (shift g x)(h) = x(g⁻¹ * h). This is the standard left shift action of G on A^G. We define it as a plain function rather than a MulAction instance to avoid conflict with the pointwise Pi.instMulAction.

Equations

• GottschalkSurjunctivity.shift G g x h = x (g⁻¹ * h)

@[simp]

theorem GottschalkSurjunctivity.shift_apply (G : Type u_1) [Group G] {A : Type u_2} (g : G) (x : G → A) (h : G) : shift G g x h = x (g⁻¹ * h)

@[simp]

theorem GottschalkSurjunctivity.shift_one (G : Type u_1) [Group G] {A : Type u_2} (x : G → A) : shift G 1 x = x

theorem GottschalkSurjunctivity.shift_mul (G : Type u_1) [Group G] {A : Type u_2} (g₁ g₂ : G) (x : G → A) : shift G (g₁ * g₂) x = shift G g₁ (shift G g₂ x)

def GottschalkSurjunctivity.IsShiftEquivariant (G : Type u_1) [Group G] {A : Type u_2} (τ : (G → A) → G → A) : Prop

A map τ : (G → A) → (G → A) is equivariant (with respect to the left shift action) if τ(shift g x) = shift g (τ x) for all g : G and x : G → A.

Equations

• GottschalkSurjunctivity.IsShiftEquivariant G τ = ∀ (g : G) (x : G → A), τ (GottschalkSurjunctivity.shift G g x) = GottschalkSurjunctivity.shift G g (τ x)

def GottschalkSurjunctivity.IsSurjunctive (G : Type u_1) [Group G] : Prop

A group G is surjunctive if for every finite nonempty type A, every injective, continuous, shift-equivariant map (G → A) → (G → A) is also surjective.

Continuity is with respect to the product topology on G → A where A carries the discrete topology.

Equations

• One or more equations did not get rendered due to their size.

theorem GottschalkSurjunctivity.gottschalk_surjunctivity_conjecture (G : Type) [Group G] : IsSurjunctive G

Gottschalk's surjunctivity conjecture (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.

theorem GottschalkSurjunctivity.isSurjunctive_of_finite (G : Type) [Group G] [Finite G] : IsSurjunctive G

Every finite group is surjunctive. This is a classical result: an injective endomorphism of a finite set is surjective.