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

TL;DR

This paper establishes a priori bounds for a specific class of infinitely renormalizable quadratic polynomials with complex combinatorics, introducing new geometric tools to handle convergence issues near the Mandelbrot set cusp.

Contribution

It develops uniform thin-thick decompositions for bordered Riemann surfaces to prove bounds for 'elephant eye' combinatorics in quadratic dynamics.

Findings

Proved a priori bounds for 'elephant eye' combinatorics.

Introduced a new geometric method: uniform thin-thick decompositions.

Handled convergence of M-copies to the Mandelbrot cusp.

Abstract

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.