Dynamics in One Complex Variable

Milnor, John

doi:10.1007/978-3-663-08092-3

Cited by 842 publications

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“…A point z 0 ∈ P 1 (C) is said to be exceptional if the set {ϕ (−n) (z 0 ) : n ∈ N} of backward iterates of z 0 is finite. It is known (see [23]) that there are at most 2 exceptional points for ϕ in P 1 (C). Proofs of the following theorem can be found in [21], [17], and [18].…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Baker¹,

Rumely²

2006

Ann. inst. Fourier

101

154

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Given a dynamical system associated to a rational function ϕ(T ) on P 1 of degree at least 2 with coefficients in a number field k, we show that for each place v of k, there is a unique probability measure µ ϕ,v on the Berkovich space P 1 Berk,v /C v such that if {z n } is a sequence of points in P 1 (k) whose ϕ-canonical heights tend to zero, then the z n 's and their Galois conjugates are equidistributed with respect to µ ϕ,v . In the archimedean case, µ ϕ,v coincides with the well-known canonical measure associated to ϕ. This theorem generalizes a result of Baker-Hsia [BH] when ϕ(z) is a polynomial.The proof uses a polynomial lift F (x, y) = (F 1 (x, y), F 2 (x, y)) of ϕ to construct a twovariable Arakelov-Green's function g ϕ,v (x, y) for each v. The measure µ ϕ,v is obtained by taking the Berkovich space Laplacian of g ϕ,v (x, y), using a theory developed in [RB]. The other ingredients in the proof are (i) a potential-theoretic energy minimization principle which says that g ϕ,v (x, y) dν(x)dν(y) is uniquely minimized over all probability measures ν on P 1 Berk,v when ν = µ ϕ,v , and (ii) a formula for homogeneous transfinite diameter of the v-adic filled Julia set K F,v ⊂ C 2 v in terms of the resultant Res(F ) of F 1 and F 2 . The resultant formula, which generalizes a formula of DeMarco [DeM], is proved using results from [RLV] about Chinburg's sectional capacity. A consequence of the resultant formula is that the product of the homogeneous transfinite diameters over all places is 1.

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Baker¹,

Rumely²

2006

Ann. inst. Fourier

101

154

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“…Conversely, every periodic point is a landing point of finitely many external rays with rational angles (although this is non-trivial to prove, see e.g. [8]). …”

Section: The Yoccoz Puzzlementioning

confidence: 99%

The dual nest for degenerate Yoccoz puzzles

Aspenberg

2009

Conform. Geom. Dyn.

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Abstract. The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the BrannerHubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows us to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.

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“…Lemma 1.5 (Douady, Sullivan) Proof. We reproduce the proof of non-local connectivity given by Douady and Sullivan in the case of polynomials with Cremer points (see [Mi1,Su]). We assume for contradiction that the Julia set is locally connected so that in particular B(p) is simply connected and the boundary of B(p) is also locally connected.…”

Section: Theoremmentioning

confidence: 99%