Non-degenerate near-parabolic renormalization

Abstract

Invariant classes under parabolic and near-parabolic renormalization have proved extremely useful for studying the dynamics of polynomials. The first such class was introduced by Inou-Shishikura to study quadratic polynomials; their argument has been extended to the unicritical cubic case by Yang and the general unicritical case by Chéritat. However, all of these classes are only applicable to maps which have a fixed point with multiplier close to one, though it is well-known that similar phenomena occur when the multiplier is close to any root of unity. In this paper we define the parabolic and near-parabolic renormalization operators in the general setting and construct invariant classes. In the general setting we can observe a new phenomenon: the multiplier may be close to several roots unity. In this case, we show how to directly relate the different near-parabolic renormalizations that arise.

Figures (6)

• Figure 1: A parabolic flower for $f$ with tilt $t$ and $p/q=-1/3$.
• Figure 2: A $(-1/3, +)$-near-parabolic flower for $g$ with tilt $t$. The spiraling of the petals around zero is controlled by Proposition \ref{['prop:petals perturbed']}. The points entering and exiting $P_0=P_1$ are shown in lighter and darker gray respectively.
• Figure 3: The parabolic renormalization $\mathcal{R}_0^\pm G$ when $d = 2$.
• Figure 4: The relationship between flowers as in Proposition \ref{['prop:comparing petals']}. Note that the petals should be spiraling around the fixed points; we omit this detail for clarity of the image.
• Figure 5: Construction of a parabolic flower.

Theorems & Definitions (88)

• Remark 1.1
• Theorem 1.2
• Proposition 1.3
• Proposition 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Proposition 1.8
• Proposition 1.9
• proof