Theorems · Theorem · dynamical systems

AddAction.isTopologicallyTransitive_iff_dense_of_preimage_invariant

∀ (M : Type u_1) {α : Type u_2} [inst : TopologicalSpace α] [inst_1 : AddMonoid M] [inst_2 : AddAction M α]
  [h : ContinuousConstVAdd M α],
  AddAction.IsTopologicallyTransitive M α ↔
    ∀ {U : Set α}, IsOpen U → U.Nonempty → (∀ (m : M), (fun x => m +ᵥ x) ⁻¹' U ⊆ U) → Dense U

An action of an additive monoid M on α that is continuous in the second argument is topologically transitive if and only if any nonempty open subset U of α with (m +ᵥ ·) ⁻¹' U ⊆ U for all m : M is dense in α.

Defined in

Mathlib.Dynamics.Transitive

Cited by

0 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddMonoidAddActionContinuousConstVAdd