Local Connectivity of the Julia Set of Real Polynomials

Real bounds, ergodicity and negative Schwarzian for multimodal maps

“…To prove this we use an argument similar to the one given on pp. 482-483 of [9]. More precisely, assume by contradiction that y accumulates onto a parabolic or fixed point a.…”

“…Claim 4: If y and C = C(y) are as in Claim 3 and so that y is not eventually mapped onto a critical point, then c ∈ ω(c ) for any c, c ∈ C. This follows from the argument given in Proposition 3.3 of [9] (we use here that Y does not have full measure in any interval). To prove this, let us first assume by contradiction that for some c ∈ C there exists one or more critical points c ∈ C such that c / ∈ ω(c ).…”

On the measurable dynamics of real rational functions

“…In this section, we recall two notions: one is 'a real box mapping' which we used in [26]; and the other is 'a quasi-polynomial-like mapping', used by Levin and van Strien in [12], which turns out to be the complex counterpart of the first notion.…”

“…We allow intersection of these domains U j i in order to obtain an advantage in constructing extensions. Note that we even allow the real traces of U j i to intersect each other, which was excluded in [12].…”

Ergodicity of conformal measures for unimodal polynomials

“…There exists a conformal mapping between the annulus A = V \ U and a standard annulus {z : 1 < |z| < r} for some r > 1. The modulus of A is defined as 1 π log r. We say that an infinitely renormalizable polynomial f has a priori bounds if there exists an > 0 such that mod(R m (f )) > , where R m (f) is the m th renormalization of f , for infinitely many m. According to [GS], [LvS95] and [LyuY95], all infinitely renormalizable real unimodal polynomials have a priori bounds. By unimodal polynomial, we mean an even degree polynomial with just one critical point.…”