Weixiao Shen: Density of Axiom A for real polynomials with real critical points (Mini Course)

I shall discuss a recent result (obtained in joint work with Oleg Kozlovski and Sebastian van Strien) on complex iteration of real polynomials with real critical points, where we prove the following rigidity result.

Given any two polynomials $f$ and $\tilde{f}$ with real coefficients and real critical points, if they are topologically conjugate on the real line (+some mild technical assumptions), then they are qc conjugate on the complex plane.

We shall explain the proof of this result as well as its main consequence: AXIOM A MAPS ARE DENSE IN REAL ONE-DIMENSIONAL DYNAMICS. In the first talk I shall review some backgrounds and discuss the relation between our work and previous ones in the literature. In the second talk we shall explain the strategy of the proof of the rigidity result.