mathlib3 documentation

dynamics.ergodic.add_circle

Ergodic maps of the additive circle #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains proofs of ergodicity for maps of the additive circle.

Main definitions: #

If a null-measurable subset of the circle is almost invariant under rotation by a family of rational angles with denominators tending to infinity, then it must be almost empty or almost full.

theorem add_circle.ergodic_zsmul {T : } [hT : fact (0 < T)] {n : } (hn : 1 < |n|) :
theorem add_circle.ergodic_nsmul {T : } [hT : fact (0 < T)] {n : } (hn : 1 < n) :
theorem add_circle.ergodic_zsmul_add {T : } [hT : fact (0 < T)] (x : add_circle T) {n : } (h : 1 < |n|) :
theorem add_circle.ergodic_nsmul_add {T : } [hT : fact (0 < T)] (x : add_circle T) {n : } (h : 1 < n) :