Geometry of $Q$-recurrent maps

Abstract

Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to the Julia set of $Q$. We use this fact to compute analytic invariants of the nest of $f_c$, to give a complete characterization of complex quadratic Fibonacci maps and to obtain a new auto-similarity result on the Mandelbrot set.

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References in corpus (1)

• Quadratic polynomials and combinatorics of the principal nest

Cited by in corpus (1)

• Quadratic polynomials and combinatorics of the principal nest