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On Lattès maps. (English) Zbl 1235.37015
Hjorth, Paul G. (ed.) et al., Dynamics on the Riemann sphere. A Bodil Branner Festschrift. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-011-6/hbk). 9-43 (2006).
From the text: This is an exposition of the 1918 paper of S. Lattès [C. R. Acad. Sci. Paris 166, 26–28 (1918; JFM 46.0522.01)], together with its historical antecedents, and its modern formulations and applications.
In 1918, some months before his death of typhoid fever, Samuel Lattès published a brief paper describing an extremely interesting class of rational maps. Similar examples had been described by Schröder almost fifty years earlier, but Lattès’ name has become firmly attached to these maps, which play a basic role as exceptional examples in the holomorphic dynamics literature.
His starting point was the “Poincaré function” $$\theta: \mathbb C\to\hat \mathbb C$$ associated with a repelling fixed point $$z_0 = f(z_0)$$ of a rational function $$f:\hat \mathbb C \to \hat \mathbb C$$. This can be described as the inverse of the Koenigs linearization around $$z_0$$, extended to a globally defined meromorphic function. Assuming for convenience that $$z_0 = \infty$$, it is characterized by the identity $$f(\theta(t)) = \theta(\mu t)$$ for all complex numbers $$t$$, with $$\theta(0) = z_0$$, normalized by the condition that $$\theta(0) = 1$$. Here $$\mu = f(z_0)$$ is the multiplier at $$z_0$$, with $$|\mu| > 1$$. This Poincaré function can be computed explicitly by the formula $$\theta(t) = \lim_{n\to\infty} f\circ n(z_0 + t/\mu n)$$.
We will expand on this idea in the following sections. Section 2 will introduce rational maps which are finite quotients of affine maps. (These are more commonly described in the literature as rational maps with flat orbifold metric – see §4.) They can be classified into power maps, Chebyshev maps, and Lattès maps according as the Julia set is a circle, a line or circle segment, or the entire Riemann sphere. These maps will be studied in Sections 3 through 5, concentrating on the Lattès case. Section 6 will describe the history of these ideas before Lattès; and Section 7 will describe some of the developments since his time. Finally, Section 8 will describe a number of concrete examples
For the entire collection see [Zbl 1081.00011].

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 37-03 History of dynamical systems and ergodic theory 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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