[Paper Review] Hyperbolicity is dense in the real quadratic family

Grzegorz Świątek|arXiv (Cornell University)|Jul 6, 1992

Mathematical Dynamics and Fractals8 references22 citations

TL;DR

This paper proves that hyperbolic maps are dense in the real quadratic family $ f_a(x) = ax(1-x) $ by establishing that topological conjugacy between non-hyperbolic real quadratic polynomials with bounded critical orbits and aperiodic kneading sequences implies quasiconformal conjugacy. The key result is that such conjugacy classes are singletons in the parameter space, which, combined with quasiconformal rigidity, implies density of hyperbolicity in the real quadratic family.

ABSTRACT

It is shown that for non-hyperbolic real quadratic polynomials topological and quasisymmetric 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.

Motivation & Objective

• To establish that topological conjugacy between non-hyperbolic real quadratic polynomials with bounded forward critical orbits and aperiodic kneading sequences implies quasiconformal conjugacy.
• To prove that quasiconformal conjugacy classes in the quadratic family are either single points or open sets, leveraging quasiconformal rigidity.
• To show that the set of parameters with aperiodic kneading sequences is both closed and locally finite, implying at most one such map per kneading sequence.
• To resolve the Dense Hyperbolicity Conjecture by eliminating the possibility of intervals filled with non-hyperbolic maps without attracting cycles.
• To confirm Milnor and Thurston's conjecture on monotonicity of the kneading invariant by showing that hyperbolic parameters are dense in the parameter space.

Proposed method

• Reduces the main problem to proving quasisymmetric conjugacy on the real line between two real quadratic polynomials with the same aperiodic kneading sequence and bounded critical orbits.
• Applies the pull-back construction from quasisymmetric conjugacy on the real line to quasiconformal conjugacy in the complex plane, using techniques from [17] and [18].
• Uses the inducing process on the real line to analyze the dynamics and construct annuli with controlled conformal moduli.
• Employs normalized symbols and estimates on conformal moduli $ s_i $, $ \lambda_i $, and separation indices $ \beta $ to control distortion in the pull-back process.
• Applies superadditivity of conformal moduli to ensure that annuli with modulus at least $ \beta_0/2 $ exist and pull back to annuli with modulus $ \geq \beta_0/4 $, maintaining control over distortion.
• Uses induction to show that the separation index $ \beta_0 $ remains valid through successive pull-backs, even after non-close returns, ensuring quasiconformal control.

Experimental results

Research questions

• RQ1 Are topological conjugacy classes of non-hyperbolic real quadratic polynomials with aperiodic kneading sequences and bounded critical orbits equal to quasiconformal conjugacy classes?
• RQ2 Can quasiconformal rigidity be used to show that each such conjugacy class contains at most one map in the real quadratic family?
• RQ3 Does the absence of attracting or indifferent cycles in a real quadratic polynomial imply that topological conjugacy implies quasiconformal conjugacy?
• RQ4 Is it possible for an interval in the parameter space to be filled with non-hyperbolic maps having periodic kneading sequences but no attracting cycles?
• RQ5 Does the density of hyperbolic maps in the real quadratic family follow from the equivalence of topological and quasiconformal conjugacy classes in the non-hyperbolic case?

Key findings

• Topological conjugacy between two non-hyperbolic real quadratic polynomials with bounded forward critical orbits and aperiodic kneading sequences implies quasiconformal conjugacy.
• Each quasiconformal conjugacy class in the quadratic family contains at most one representative, due to quasiconformal rigidity.
• The set of parameters with a given aperiodic kneading sequence is both closed and locally finite, so it contains at most one such map.
• The absence of attracting or indifferent cycles in a real quadratic polynomial implies that its topological conjugacy class is either a singleton or open in the parameter space.
• The Dense Hyperbolicity Theorem holds: hyperbolic maps are dense in the real quadratic family $ f_a(x) = ax(1-x) $, as there are no intervals of non-hyperbolic maps without attracting cycles.
• The result confirms Milnor and Thurston’s conjecture that the kneading sequence is strictly increasing unless periodic, since otherwise an interval of aperiodic kneading sequences would contradict the theorem.