Cohomological equations and invariant distributions for minimal circle diffeomorphisms

Abstract

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real C ∞ -coboundaries of such a diffeomorphism is closed in C ∞ (T) if and only if its rotation number is Diophantine.

Kinematic expansive flows

“…In this section we apply the results of [3] to study smooth kinematic expansive suspensions of irrational rotations. THEOREM 4.11.…”

“…there is δ > 0 such that if 0 < dist(x, y) < δ, then T (x) = T (y); and (3) l is homeomorphic to a subset of R.Proof. (1 → 2) The return time map T making the suspension of the identity map of l kinematic expansive, has to be locally injective by Proposition 4.1(3).…”

On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

“…Theorem A allows to show an improved version of the Denjoy-Koksma inequality for C 1 test functions, thus extending Corollary C of [1] (valid for diffeomorphisms of class C 11 ) to C 1+bv diffeomorphisms. We omit the proof since it follows the very same lines of that of [1]. Indeed, the only new tool needed in [1] was the absence of invariant 1-distributions other than the invariant measure.…”

“…We omit the proof since it follows the very same lines of that of [1]. Indeed, the only new tool needed in [1] was the absence of invariant 1-distributions other than the invariant measure.…”

On manifolds supporting distributionally uniquely ergodic diffeomorphisms

“…Ergodic translations on tori are the archetypical examples of DUE diffeomorphisms. Recently, the first and third authors showed in [AK11] that every smooth circle diffeomorphism with irrational rotation number is also DUE.…”