Trees and the dynamics of polynomials

DeMarco, Laura; McMullen, Curtis T.

doi:10.24033/asens.2070

Cited by 39 publications

(48 citation statements)

References 27 publications

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“…For the special case where a(t) is a critical point of f t , the following proposition was established in [DM,Proposition 10.4] (and a version for rational functions was proved in [DF, Theorem 2.5]). We give a different proof, appealing to properties of the function-field height of f t .…”

Section: Activity and Normal Familiesmentioning

confidence: 99%

“…The polynomial growth of the coefficients of f t implies that M(f t ) grows logarithmically in t. Indeed, by passing to a finite cover of the punctured disk {|t| > R} for some R 0, we may assume that the critical points of f t are holomorphic functions of t. Applying [DM,Proposition 10.4], which uses standard distortion estimates for univalent functions, we conclude that M(f t ) = e log |t| + O(1) as t → ∞ for some e > 0.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

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Special Curves and Postcritically Finite Polynomials

Baker¹,

Marco²

2013

Forum math. Pi

106

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We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on P 1 containing a Zariski-dense subset of postcritically finite maps.2010 Mathematics Subject Classification: 37F45 (primary); 11G50, 30C10 (secondary) OverviewIn this paper we address the question 'Which algebraic subvarieties of the moduli space M d of degree-d rational maps contain a Zariski-dense set of special points?' Here, a special point is the conjugacy class of a postcritically finite map f : P 1 → P 1 ; that is, every critical point of f has finite forward orbit under iteration. Postcritically finite (PCF) maps play an important role in complex dynamics, and in recent years there has been an explosion of work around them. It has been known since the foundational work of Thurston based on Teichmüller theory that PCF maps come in two flavors, the flexible Lattès maps (associated with elliptic curves and arising in one-dimensional families) and the rest (which are rigid). The rigid PCF maps form a countable Zariski-dense subset of the moduli space M d . From an arithmetic point of view, PCF maps are in several ways similar to elliptic curves with complex multiplication, and one can therefore view the above question as a dynamical analog of the André-Oort conjecture in arithmetic geometry. We formulate a conjectural answer to the above question: the special subvarieties V having a dense set of special points should M. Baker and L. DeMarco 2 be those for which the number of dynamically independent critical points does not exceed the dimension of V.As evidence for our general conjecture, we study rational curves within the parameter space of critically marked, monic, centered, degree-d polynomials. Our main result provides an explicit description of the polynomially parameterized rational curves that are special: they are those for which there is exactly one active critical orbit, up to polynomial symmetries. (We in fact prove a more general result about marked but not necessarily critical points that are simultaneously preperiodic.) We provide examples showing that the 'up to symmetries' condition is necessary, and we illustrate how one can check that a given curve is special. We also study the family of curves Per 1 (λ) inside the space of cubic polynomials. First introduced by Milnor, Per 1 (λ) is defined as the set of maps with a fixed point of multiplier λ. The curve Per 1 (0), defined by the condition that one critical point is fixed, is special, and we prove that Per 1 (λ) is not special for all λ = 0.The proofs of these results rely on several ingredients, including: (1) an arithmetic equidistr...

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Section: Activity and Normal Familiesmentioning

confidence: 99%

Section: Preliminary Definitionsmentioning

confidence: 99%

Special Curves and Postcritically Finite Polynomials

Baker¹,

Marco²

2013

Forum math. Pi

106

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“…The measure µ 0 is equal to the residual equilibrium measure for the induced rational map f : P 1 by Jan Kiwi in his work on cubic polynomials and quadratic rational maps; see [13,14] and [3]. A closely related construction, viewing degenerations of polynomial maps as actions on trees, can be seen in [7]. Charles Favre has recently constructed a compactification of the space of rational maps, where the boundary points are rational maps on a Berkovich P 1 [10].…”

Section: Introductionmentioning

confidence: 99%

Degenerations of Complex Dynamical Systems

Marco¹,

Faber²

2014

Forum math. Sigma

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We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphereĈ must be a countable sum of atoms. For a one-parameter family f t of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit onĈ as the family degenerates. The family f t may be viewed as a single rational function on the Berkovich projective line P 1 L over the completion of the field of formal Puiseux series in t, and the limiting measure onĈ is the 'residual measure' associated with the equilibrium measure on P 1 L . For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure on P 1 L .

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“…[DM,Theorem 1.3] In the geometric topology, the set of normalized polynomial trees in T d is compact.…”

Section: Trees and Polynomialsmentioning

confidence: 99%

Finiteness for degenerate polynomials

DeMarco¹

2008

Holomorphic Dynamics and Renormalization

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Abstract. Let MP d denote the space of polynomials f : C → C of degree d ≥ 2, modulo conjugation by Aut(C). Using properties of polynomial trees (as introduced in [DM]), we show that if fn is a divergent sequence of polynomials in MP d , then any subsequential limit of the measures of maximal entropy m(fn) will have finite support. With similar techniques, we observe that the iteration maps {MP d MP d n : n ≥ 1} between GIT-compactifications can be resolved simultaneously with only finitely many blow-ups of MP d .

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Trees and the dynamics of polynomials

Cited by 39 publications

References 27 publications

Special Curves and Postcritically Finite Polynomials

Special Curves and Postcritically Finite Polynomials

Degenerations of Complex Dynamical Systems

Finiteness for degenerate polynomials

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