On the measurable dynamics of real rational functions

Abstract

Abstract. Let f be a real rational function with all critical points on the extended real axis and of even order. Then: (1) f carries no invariant line field on the Julia set unless it is doubly covered by an integral torus endomorphism (a Lattés example); and (2) f |J (f ) has only finitely many ergodic components.

Density of hyperbolicity in dimension one

“…In the case of minimal ω(c 0 ) the existence of the box mapping is proved in [She04], and the absence of an invariant line field follows from the same argument in Sections 6 and 7 of [She03]. So we only have to prove the nonminimal case.…”

Phase space universality for multimodal maps

“…Shen [Sh1] proved that any real polynomial with real critical points does not support invariant line fields in it Julia set. The above situation is not included, in general, in Shen's result, but we can expect the same result (probably with a very similar proof) for compositions of real polynomials with real critical points.…”

Solenoidal attractors with bounded combinatorics are shy

“…Then the cocycle L is uniformly expanding, that is, there is C > 0 and θ 2 > 1 such that for every v ∈ B nor (U ) and F ∈ Ω n,p we have By (37) we have that v ∈ E h F . So L is uniformly expanding.…”