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Relation to Complex Dynamics

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Rotation Sets and Complex Dynamics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2214))

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Abstract

In this chapter we outline how rotation sets occur in the dynamical study of complex polynomial maps. Special attention is paid to the relation with the dynamics of complex quadratic and cubic polynomials. This link provides a geometric realization of rotation sets under m d , whose abstract theory was developed in the previous chapters.

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Notes

  1. 1.

    It is conjectured that ∂Δ is a Jordan curve containing a critical point for almost every rotation number θ. This has been proved in the quadratic case in [25].

  2. 2.

    The limb decomposition hypothesis is believed to hold for almost every rotation number θ (and at least for θ of bounded type), but so far this has been rigorously verified only for d = 2 where the whole Julia set is known to be locally connected; see [23] and [25].

  3. 3.

    In [30] the cubics are given in the normal form

    $$\displaystyle \begin{aligned} z \mapsto e^{2\pi i \theta} z \Big( 1- \frac{1}{2} \big( 1+\frac{1}{c} \big) z + \frac{1}{3c} z^2 \Big) \qquad c \in {\mathbb C}^*, \end{aligned}$$

    with marked critical points at 1 and c. The punctured c-plane is a double-cover of the a 2-plane, branched at c = ±1. In this normalization, Γ(θ) appears as a Jordan curve passing through these branch points, and is invariant under the involution c↦1∕c.

  4. 4.

    Each parameter ± a n is the “root” of a capture component in which the (n + 1)-st iterate of one critical point hits the Siegel disk. We will not be using this fact in our presentation.

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Authors and Affiliations

  1. Department of Mathematics, Queens College of CUNY, Queens, NY, USA

    Saeed Zakeri

  2. Department of Mathematics, Graduate Center of CUNY, New York, NY, USA

    Saeed Zakeri

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  1. Saeed Zakeri
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Zakeri, S. (2018). Relation to Complex Dynamics. In: Rotation Sets and Complex Dynamics. Lecture Notes in Mathematics, vol 2214. Springer, Cham. https://doi.org/10.1007/978-3-319-78810-4_5

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