Hyperbolicity is Dense in the Real Quadratic Family

It is shown that for non-hyperbolic real quadratic polynomials topological and qua-sisymmetric conjugacy classes are the same. By quasiconformal rigidity, each class has only one representative in the quadratic family, which proves that hyperbolic maps are dense. Statement of the results. Dense Hyperbolicity Theorem In the real quadratic family f a (x) = ax(1 − x) , 0 < a ≤ 4 the mapping f a has an attracting cycle, and thus is hyperbolic on its Julia set, for an open and dense set of parameters a. What we actually prove is this: Main Theorem Let f andˆf be two real quadratic polynomials with a bounded forward critical orbit and no attracting or indifferent cycles. Then, if they are topologically conjugate, the conjugacy extends to a quasiconformal conjugacy between their analytic continuations to the complex plane. Derivation of the Dense Hyperbolicity Theorem. We show that the Main Theorem implies the Dense Hyperbolicity Theorem. Quasiconformal conjugacy classes of normalized complex quadratic polynomials are known to be either points or open (see [21].) We remind the reader, see [25], that the kneading sequence is aperiodic for a real quadratic polynomial precisely when this polynomial has no attracting or indifferent periodic orbits. Therefore, by the Main Theorem, topological conjugacy classes of real quadratic polynomials with aperiodic kneading sequences are either points or open in the space of real parameters a. On the other hand, it is an elementary observation that the set of polynomials with the same aperiodic kneading sequence in the real quadratic family is also closed. So, for every aperiodic kneading sequence there is at most one polynomial in the real quadratic family with this kneading sequence. Next, between two parameter values a 1 and a 2 for which different kneading sequences occur, there is a parameter a so that f a has a periodic kneading sequence. So, the only way the Dense Hyperbolicity Theorem could fail is if there were an interval filled with polynomials without attracting periodic orbits and yet with periodic kneading sequences. Such polynomials would all have to be parabolic (have indifferent periodic orbits). It well-known, however, by the work of [6], that there are only countably many such polynomials. The Dense Hyperbolicity Theorem follows. Consequences of the theorems. The Dense Hyperbolicity Conjecture had a long history. In a paper from 1920, see [7], Fatou expressed the belief that " general " (generic in today's language?) rational …