Weixiao Shen

On the measurable dynamics of real rational functions

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 Lattes example); and (2) f|J(f) has only finitely many ergodic components.

On stochastic stability of expanding circle maps with neutral fixed points

It is well known that the Manneville–Pomeau map with a parabolic fixed point of the form is stochastically stable for α ≥ 1 and the limiting measure is the Dirac measure at the fixed point. In this paper, we show that if α ∈ (0, 1), then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.