[Paper Review] Remarks on the Dynamic of the Ruelle Operator and invariant differentials

Peter Makienko|ArXiv.org|Apr 29, 2008

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper investigates the dynamics of the Ruelle operator on spaces of differentials for rational maps, establishing necessary and sufficient conditions—via convergence of measure sequences and behavior of Ruelle-Poincaré series—for the nonexistence of invariant conformal structures on the Julia set. The key contribution is a proof that, except for Lattés maps, no invariant regular (including augmented meromorphic) differentials exist under postcritical set conditions, extending prior results on integrable differentials.

ABSTRACT

Let $ R $ be a rational map. We are interesting in the dynamic of the Ruelle operator on suitable spaces of differentials. In particular the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence of invariant conformal structures on $ J(R) $ are obtained.

Motivation & Objective

• To characterize the existence of invariant conformal structures on the Julia set of rational maps using the dynamics of the Ruelle operator.
• To extend previous results on invariant integrable differentials to the broader class of regular differentials, including augmented meromorphic ones.
• To establish necessary and sufficient conditions—based on convergence of measure sequences and Ruelle-Poincaré series—for the absence of invariant conformal structures and for the Julia set to have zero Lebesgue measure.
• To reformulate additional conditions from prior works into equivalent dynamical criteria involving the Ruelle operator's modulus and spectral behavior.
• To investigate the spectrum of the Ruelle and Beltrami operators, particularly in relation to J-stability and hyperbolicity.

Proposed method

• The Ruelle operator $ R^* $ is defined on $ L_1(\overline{\mathbb{C}}) $ and $ L_p(\overline{\mathbb{C}}) $, acting on differentials via pullback through inverse branches of the rational map $ R $.
• The modulus of the Ruelle operator $ |R^*| $ is used to analyze the spectral behavior and convergence of sequences of measures related to invariant structures.
• A quasiconformal map $ f_\mu $ associated with a candidate invariant conformal structure $ \mu $ is constructed, and conditions are derived under which a related map $ h $ can be supported on the Fatou set with the same infinitesimal deformation.
• The proof relies on contradiction: assuming the existence of a nontrivial invariant regular differential leads to a finite forward orbit of a critical point, violating the assumption of infinite orbit.
• The authors use the generalized Sullivan conjecture and properties of Poincaré series to connect the nonexistence of invariant differentials to the absence of nontrivial fixed points of the Beltrami operator.
• Spectral analysis of $ R^* $ on $ L_1 $ shows the spectrum is the closed unit disk, and the structure of eigenfunctions is used to derive contradictions under infinite postcritical orbits.

Experimental results

Research questions

• RQ1 Under what conditions does the Ruelle operator's dynamics imply the nonexistence of invariant conformal structures on the Julia set?
• RQ2 How do the convergence of sequences of measures and the behavior of Ruelle-Poincaré series relate to the absence of invariant conformal structures?
• RQ3 Is the absence of invariant regular differentials (beyond integrable ones) equivalent to the map being non-Lattés, under postcritical set conditions?
• RQ4 Can the spectrum of the Ruelle operator on the closure of the linear span of Poincaré series functions be finite only for postcritically finite maps?
• RQ5 Does J-stability of $ R^k $ in its Hurwitz class imply that $ R^n $ is J-stable for $ n > k $, and what does this imply for the Beltrami operator's point spectrum?

Key findings

• The Ruelle operator $ R^* $ on $ L_1(\overline{\mathbb{C}}) $ has the closed unit disk as its spectrum, with every interior point being an eigenvalue.
• For any rational map $ R $, the absence of nontrivial invariant regular differentials (including augmented meromorphic ones) is equivalent to $ R $ not being a Lattés map, under the same postcritical set assumptions as in [Mak2].
• The necessary and sufficient condition for the Julia set to have Lebesgue measure zero is the convergence of a specific sequence of measures related to the Ruelle operator's action on Poincaré series.
• The nonexistence of invariant conformal structures on $ J(R) $ is equivalent to the convergence of the sequence of measures $ \mu_n $ defined by $ \mu_n = (R^*)^n(\tau_a) $, where $ \tau_a $ is a Poincaré series function.
• The Ruelle-Poincaré series' common behavior—specifically, absolute convergence and decay—serves as a sufficient condition for the nonexistence of invariant conformal structures, generalizing earlier results.
• The proof shows that if a nontrivial invariant regular differential exists, then a critical point with infinite forward orbit must have finite forward orbit, contradicting the assumption, thus proving nonexistence.