Brennan Conjecture for Basin of Attraction at Infinity

Abstract: This paper investigates the Brennan Conjecture for domains $Ω$ that arise as basins of attraction of a polynomial. We extend the result of Baranski, Volberg, and Zdunik to a broader class of polynomials. We prove that for any monic polynomial of degree $m$ with sufficiently small non-leading coefficients, Brennan Conjecture holds for its basin of attraction.

• The paper extends the Brennan Conjecture to basins of attraction at infinity for higher-degree monic polynomials with small non-leading coefficients.
• It employs Koebe’s Distortion Theorem and recursive growth estimates to convert the integral convergence problem into a geometric-series criterion.
• The analysis provides explicit bounds on the allowable coefficients and demonstrates convergence even for exponents slightly above 4 in symmetric cases.

Brennan Conjecture for Basins of Attraction at Infinity: Summary and Analysis

Introduction and Background

The paper "Brennan Conjecture for Basin of Attraction at Infinity" (2604.12240) examines the longstanding Brennan Conjecture in the context of basins of attraction at infinity for polynomials. The conjecture concerns the range of exponents $p$ such that the integral

$\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$

where $F:\Omega\to\mathbb{D}$ is a conformal bijection from a simply-connected domain $\Omega$ to the unit disk. The conjecture asserts finiteness of the integral for all $p \in (\frac{4}{3},4)$. Numerical bounds have been gradually improved: Hedenmalm and Shimorin gave convergence for $p<3.75$, building on earlier results.

A central question of the paper is whether the Brennan Conjecture holds for domains $\Omega$ that are the basins of attraction at infinity for polynomials $f:\mathbb{C}\to\mathbb{C}$. The work of Barański, Volberg, and Zdunik established this for quadratic ($m=2$) polynomials. This paper generalizes to higher-degree polynomials with "small" non-leading coefficients.

Main Results

The author's key claims are as follows:

1. Extension to Monic Polynomials: For any monic polynomial $f(z)=z^m + a_0$ of degree $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$0 with $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$1 simply-connected, the Brennan Conjecture holds on $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$2 (Theorem: "pure generalization of BVZ").
2. Perturbative Generalization: For general monic polynomials $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$3, if the non-leading coefficients satisfy $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$4 for all $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$5 (where $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$6 is explicit but small), and $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$7 is simply-connected, then the Brennan Conjecture holds on $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$8.

The analysis particularly addresses the case of higher degrees $\int_{\Omega}|F'(z)|^p\, dA(z)<+\infty,$9, resolving the practical difficulties left open after $F:\Omega\to\mathbb{D}$0 and substantially extending the class of polynomial dynamical domains for which the conjecture is resolved.

Methodology

The core technical content includes a reduction of the Brennan integral to a geometric-series criterion on iterated derivatives evaluated at backward orbits of a basepoint. The author uses:

• Koebe's Distortion Theorem to locally control the variation of the derivative in small regions partitioned around iterated preimages.
• Change of Variable via the Inverse Map: The integral on $F:\Omega\to\mathbb{D}$1 is converted, using the inverse function, into an integral over the disk and further to a sum over preimages of orbits under the polynomial dynamics.
• Recursive Growth Estimates: By relating the sum $F:\Omega\to\mathbb{D}$2 over the $F:\Omega\to\mathbb{D}$3-th backward orbit points to $F:\Omega\to\mathbb{D}$4, and bounding the growth via the explicit form of $F:\Omega\to\mathbb{D}$5 and its iterates.
• Critical Point Control: Via results of Carleson and others, provides sufficient conditions for simply-connectedness of $F:\Omega\to\mathbb{D}$6 in terms of the locations of critical points, and ensures that small perturbations of $F:\Omega\to\mathbb{D}$7 do not change the qualitative topology of $F:\Omega\to\mathbb{D}$8.
• Continuity and Perturbation Theory in Roots and Preimages: Relates the effect of small coefficients to geometric properties of preimages, showing key bounds persist under sufficiently small perturbations.

The stepwise estimation for $F:\Omega\to\mathbb{D}$9 exploits the rotational symmetry of monic polynomials, but for perturbed cases, continuity properties and explicit estimates underpin the claims.

Strong Numerical and Structural Statements

The paper provides explicit bounds on the permissible size of the non-leading coefficients $\Omega$0 as a function of degree, ensuring the geometric recurrence structure is not destroyed. The result is specific: for any $\Omega$1, there exists $\Omega$2 such that the conjecture holds for all polynomials within the specified coefficient domain.

For large $\Omega$3, it is explicitly shown that the key geometric sum $\Omega$4 is strictly less than $\Omega$5, ensuring the rapid decay required for integral convergence. In fact, the proof technique allows for exponents $\Omega$6 slightly greater than $\Omega$7 to yield a finite integral in certain symmetric cases, which is emphasized as a technically significant point, albeit not a contradiction of the conjecture's range.

Implications and Future Directions

The extension of the Brennan Conjecture to basins of attraction for higher-degree polynomials, especially in the setting of small non-leading coefficients, provides a new class of non-trivial simply-connected domains where the conjecture is verified. This advances the interface between geometric function theory and holomorphic dynamics, as these basins are canonical examples of dynamically-generated domains with rich boundary structure.

On a theoretical level, the methods illustrate powerful connections between conformal mapping, polynomial dynamics (iteration theory), and geometric measure estimates for conformal derivatives. The perturbative approach creates potential pathways for further generalizations, possibly to rational functions or more general dynamical domains.

Practically, these results can influence numerical and analytical techniques in conformal mapping, especially for domains arising in iterative processes, as well as providing motivation to examine the dynamics of critical points in more detail for global analytic properties.

Open directions include the extension to broader classes of dynamical systems (e.g., non-monic or rational maps), optimization of the coefficient bounds, and deeper investigation of the fine structure of boundaries in $\Omega$8 under general perturbations.

Conclusion

This paper establishes the validity of the Brennan Conjecture for basins of attraction at infinity for monic polynomials of arbitrary degree with sufficiently small non-leading coefficients, thereby extending previous work beyond the quadratic case. By blending conformal geometry, complex dynamics, and function-theoretic techniques, the author provides both a methodological and substantive contribution to the study of integral means of derivatives on dynamically-generated domains and opens up several avenues for further mathematical exploration.