A priori bounds for some infinitely renormalizable quadratic: IV. Elephant Eyes

In this paper we prove a priori bounds for an ``elephant eye’’ combinatorics. Little $M$-copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.

💡 Research Summary

The paper establishes a priori bounds for a class of infinitely renormalizable quadratic polynomials whose combinatorics are called “elephant eyes”. Unlike previously studied renormalizable parameters, which involve only finitely many limbs of the Mandelbrot set, the elephant‑eye combinatorics allow the renormalization periods p and q to satisfy p = q + O(1) while p can become arbitrarily large. Moreover, the associated little copies of the Mandelbrot set (the “M‑copies”) are permitted to accumulate at a cusp of the main cardioid. This unbounded situation makes the traditional width‑based geometric control ineffective.

To overcome this difficulty the authors introduce a new geometric tool: a uniform thin‑thick decomposition for bordered Riemann surfaces. For a compact Riemann surface S with boundary components Jₖ, they assign a “local weight” W(Jₖ), defined as the conformal width of the annular covering of Jₖ minus a fixed buffer of size 2 (or zero if the width is ≤2). The total weight W(S)=∑ₖW(Jₖ) measures the overall size of the surface. The main technical result (Theorems 3.9 and 3.10) shows that the sum of the weights of all positively weighted arcs (the canonical lamination) satisfies
 W(S) − O(|χ(S)|) ≤ 2 ∑_{α∈𝔄_S} W(α) ≤ W(S),
where χ(S) is the Euler characteristic. In other words, the thin part (a collection of short hyperbolic geodesic arcs) and the thick part (the complement) together account for essentially all of the total weight, up to an error proportional to the topology of S.

The paper then develops a suite of transformation rules for these weights under holomorphic maps between bordered surfaces (Lemma 3.12, 3.13). In particular, when a map is a covering, the inequalities become equalities, while for a degree‑d map the boundary weight can increase at most by a factor d and interior gaps can lose at most a fixed constant. These rules are crucial for controlling how weights behave under the renormalization operator.

A key difficulty in the infinite‑renormalization setting is the phenomenon of “breaking”: an arc or segment in the target surface may pull back to a concatenation of several pieces in the source surface. Lemma 3.14 proves that the total weight of all breaking segments is bounded by the difference of the total weights of the source and target plus an O(p+|χ|) error term. Similar estimates hold for breaking arcs (Lemma 3.15). These bounds guarantee that, even after infinitely many renormalization steps, the accumulation of broken pieces does not cause the total weight to explode.

Section 4 translates the abstract machinery into the concrete dynamics of an even quadratic‑like map f: U′→U with connected Julia set. The map has two repelling fixed points α and β; α is the landing point of q>1 external rays that are cyclically permuted with rotation number p/q, while β is the landing point of the external 0‑ray. The rays partition the domain into 2q−1 regions (a central region, q sectors attached to α, and q symmetric sectors attached to −α). Assuming f is renormalizable with period p, there are little filled Julia sets K₀,…,K_{p−1} (with K₀ containing the critical point). The “elephant eye” combinatorics correspond to the situation where the sequence of M‑copies associated to these Kₙ converges to the cusp of the Mandelbrot set, and the periods satisfy p≈q.

Applying the thin‑thick decomposition, each little copy is split into a thin part (controlled by the hyperbolic length of a family of geodesics) and a thick part (controlled by the Euler characteristic of the corresponding puzzle piece). The thin part contributes linearly in p, while the thick part contributes only O(|χ|). Consequently the total weight of the renormalization tower remains bounded by a linear function of p, establishing the desired a priori bounds.

Finally, the authors discuss the implications for the Mandelbrot set’s local connectivity (MLC) conjecture. A priori bounds are a central hypothesis in Yoccoz‑type puzzle arguments; by proving them for the previously inaccessible “elephant eye” case, the paper extends the class of parameters for which MLC can be approached. Moreover, the uniform thin‑thick decomposition is presented as a versatile tool that may find applications beyond the specific combinatorics treated here, potentially aiding the study of other unbounded renormalization phenomena in holomorphic dynamics.