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Degree-\(d\)-invariant laminations. (English) Zbl 1452.37055

Thurston, Dylan P. (ed.), What’s next? The mathematical legacy of William P. Thurston. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 205, 259-325 (2020).
Invariant laminations were introduced by W. P. Thurston [in: Complex dynamics. Families and friends. Wellesley, MA: A K Peters. 3–109 (2009; Zbl 1185.37111)] as a tool in the study of complex dynamics.
“During the last years of his life, William P. Thurston developed a theory of degree-\(d\)-invariant laminations, a tool that he hoped would lead to what he called a qualitative picture of the dynamics of degree \(d\)-polynomials”. The first part of the present paper consists of Thurston’s unfinished manuscript, the second part is supplementary material written by other authors and based on Thurston’s Cornell seminar on these topics. “Thurston’s seminar during this period primarily focused on topics in complex dynamics. He discussed his topological characterization of rational maps on the Riemann sphere, as well as how to understand complex polynomials via topological entropy and laminations on the circle.”
From the introduction of Thurston’s manuscript: “Our overall understanding and global picture of the dynamics of degree \(d\) rational maps and even degree \(d\) complex polynomials has remained sketchy and unsatisfying. The purpose of this paper is to develop at least a sketch for a skeletal qualitative picture of degree \(d\) polynomials. We hope to contribute toward developing and clarifying the global picture of the connectedness locus for degree \(d\) polynomials, that is, the higher-dimensional analogues of the Mandelbrot set. To do this, the main tool will be the theory of degree-\(d\)-invariant laminations. We hope that developing a better picture for degree \(d\) polynomials, we will develop insights that will carry on to better understand degree \(d\) rational maps, whose global description is even more of a mystery.”
For the entire collection see [Zbl 1437.55002].

MSC:

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
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