Symmetric rigidity for circle endomorphisms having bounded geometry

Abstract

Let f f and g g be two circle endomorphisms of degree d ≥ 2 d\geq 2 such that each has bounded geometry, preserves the Lebesgue measure, and fixes 1 1 . Let h h fixing 1 1 be the topological conjugacy from f f to g g . That is, h ∘ f = g ∘ h h\circ f=g\circ h . We prove that h h is a symmetric circle homeomorphism if and only if h = I d h=Id .