Differentiable circle maps with a flat interval

Abstract

We study weakly order preserving circle maps with a flat interval, which are differentiable even on the boundary of the flat interval. We obtain estimates on the Lebesgue measure and the Hausdorff dimension of the non-wandering set. Also, a sharp transition is found from degenerate geometry to bounded geometry, depending on the degree of the singularities at the boundary of the flat interval.

On physical measures for Cherry flows

“…The length of that interval in the natural metric on the circle will be denoted by |a − b|. Following [3], let us adopt these notational conventions for the distance between the preimages of the first return function f :…”

“…We recall the following theorem proved in [3] : Theorem 2.3. Let λ 1 > 0 > λ 2 be the eigenvalues of the saddle point of φ and let f be the first return function of the reversing flow ϕ.…”

Cherry flows with non-trivial attractors

“…Moreover, g can be chosen so that on some right-sided neighborhood of a n + ǫ 2 n it is equal to h r,n ((x−(a n + ǫ 2 n )) n+1 ) for some C ∞ -diffeomorphism h r,n . Such g belongs to the class of functions with a flat interval (being J n in this case) studied in [3] (see property (5)). We notice that g m (J n ) =f m n,δ ′ (J n ) thus the orbits of g are not dense.…”

Quasi-symmetric conjugacy for circle maps with a flat interval

“…We contribute to the area of the understanding the dynamics of a class of continuous degree one circle maps with a flat interval, see [21], [19], [13], [14], [5], [4], [15], [16]. In this paper we address the problem of quasi-symmetric conjugation.…”