No hyperbolic sets in J_\infty for infinitely renormalizable quadratic polynomials

TL;DR

This paper proves that for infinitely-renormalizable quadratic polynomials, the intersection of forward orbits of small Julia sets contains no hyperbolic sets, shedding light on the complex dynamics of such systems.

Contribution

It establishes a new result showing the absence of hyperbolic sets in J_infinity for a class of quadratic polynomials, advancing understanding of their dynamical structure.

Findings

J_infinity contains no hyperbolic sets

The result applies to infinitely-renormalizable quadratic polynomials

Provides insights into the structure of Julia sets in complex dynamics

Abstract

Let f be an infinitely-renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of simple renormalizations of f. We prove that J_\infty contains no hyperbolic sets.