Topology of the regular part for infinitely renormalizable quadratic polynomials

Abstract

In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topology is rigid modulo its combinatorics.

On the topology of the inverse limit of a branched covering over a Riemann surface

“…Let us now provide one important non-cyclical case, for which one irregular point can be explicitly constructed and its signature can be explicitly computed. We refer to [16], [14], [10] and [5] for more details relevant to this case.…”

“…Recall (see, for example, Paragraph 3 in [13] and page 7 in [5]) that a quadratic polynomial f (z) = z 2 + a acting on C is called persistently recurrent, if the critical point c = 0 is not periodic and not pre-periodic and all the points of the invariant lift of ω(c) to the P.I.L. are irregular.…”

“…The subset Γ(c) of ω(c), which is the union of all the inversecritical sets with respect to c, is called the maximal inverse-critical set with respect to c.Notice that if Γ(c) is not empty, then f (Γ(c)) = Γ(c).Recall (see, for example, Paragraph 3 in[13] and page 7 in[5]) that a quadratic polynomial f (z) = z 2 + a acting on C is called persistently recurrent, if the critical point c = 0 is not periodic and not pre-periodic and all the points of the invariant lift of ω(c) to the P.I.L. are irregular.…”