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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§4. Fatou and Julia: Dynamics on the Riemann Sphere

4.1. A Brief History.* The study of iterated holomorphic mappings began in the 19th century but only came to flower in the 20th. One primary focus in the 19th century was the study of functional equations relating different known or unknown functions. Charles Babbage [1815], “An essay on the calculus of functions,” set the stage with an attempt to understand many such functional equations. In particular (on page 412), he implicitly described what we now call a semiconjugacy φ between two different functions f and g, that is a map satisfying φ o f = g o φ so that the following diagram is commutative:

It follows that any orbit x0 x1 ... under f maps to an orbit φ(xo) φ(x1) ... under g. In the ...

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