[Paper Review] Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps

Rui Gao, Weixiao Shen|arXiv (Cornell University)|Jul 11, 2012

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper studies analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps, proving that when the coupling function is a nonconstant polynomial of odd degree, the system exhibits two positive Lyapunov exponents almost everywhere and admits a unique absolutely continuous invariant probability measure. The analysis relies on conjugating the system to a uniformly expanding base map and controlling recurrence to the critical line via subhyperbolicity and analyticity properties.

ABSTRACT

We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.

Motivation & Objective

• To investigate the statistical properties of skew-products of quadratic polynomials over Misiurewicz-Thurston maps with polynomial coupling.
• To establish the existence and uniqueness of an absolutely continuous invariant probability measure (a.c.i.p.) for such systems.
• To prove that the system has two positive Lyapunov exponents almost everywhere under suitable coupling conditions.
• To extend previous results on non-uniform expansion in higher-dimensional skew-products by weakening assumptions on the base map and generalizing the coupling function.

Proposed method

• Conjugate the original skew-product map to a new system using a conjugation map $ u $, transforming the base map into a uniformly expanding map $ h_0 $ on an interval $ I_a $.
• Use the subhyperbolicity of the Misiurewicz-Thurston map $ Q_a $ to construct $ u $ such that $ u^{-1} $ and inverse branches of $ h_0 $ have analytic properties.
• Transform the system into a new skew-product $ F $ on $ I_a \times \mathbb{R} $, where the base map $ h = h_0^{m_1} $ is uniformly expanding and the coupling function $ \phi = \varphi \circ u^{-1} $ is a polynomial of odd degree.
• Control recurrence of orbits to the critical line $ y=0 $ by showing that horizontal curves become non-flat and widely spread in the $ y $-direction under iteration, using Proposition 3.1 and Lemma 4.1.
• Apply the 'Building Expansion' lemmas (Lemma 2.7) and results from non-uniform expansion theory to prove that $ F $ is non-uniformly expanding in the sense of [3].
• Verify conditions in [3, Theorem C] for existence of a.c.i.p., including decay of inverse derivative norms and recurrence control, to conclude existence and uniqueness of the invariant measure.

Experimental results

Research questions

• RQ1 Under what conditions does a skew-product of a Misiurewicz-Thurston quadratic map with a polynomial coupling function admit two positive Lyapunov exponents?
• RQ2 Does such a system possess a unique absolutely continuous invariant probability measure when the coupling function is a nonconstant polynomial of odd degree?
• RQ3 Can the assumption of odd-degree coupling be removed, or is it essential to the proof?
• RQ4 How does the conjugation to a uniformly expanding base map facilitate the control of recurrence and expansion in the skew-product system?

Key findings

• For any $ \alpha > 0 $ sufficiently small and $ m_1 \geq m_0(a) $, the skew-product map $ \mathscr{F} $ has two positive Lyapunov exponents almost everywhere.
• The system admits a unique absolutely continuous invariant probability measure with respect to Lebesgue measure.
• The vertical Lyapunov exponent satisfies $ \liminf_{n \to \infty} \frac{1}{n} \log \left| \frac{\partial f_n}{\partial y} \right| \geq \frac{\eta}{2} \log \sigma > 0 $ for almost every $ (\theta, y) $, confirming non-uniform expansion in the vertical direction.
• The conjugated system $ F $ satisfies all conditions of [3, Theorem C], ensuring the existence of an a.c.i.p. when $ \alpha $ is small enough.
• The uniqueness of the a.c.i.p. follows from topological exactness of $ F $ on its minimal forward invariant set and measure-theoretic basin properties.
• The assumption that the coupling function $ \varphi $ is of odd degree is necessary in the current proof framework, as shown in Lemma 2.6 and its discussion.