Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity

DeMarco, Laura

doi:10.1007/s00208-002-0404-7

Cited by 80 publications

(82 citation statements)

References 19 publications

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“…the homogeneous transfinite diameter of K. We will now show that when v is archimedean and K ⊂ C 2 is compact, d 0 ∞ (K) coincides with the homogeneous capacity introduced by DeMarco [12]. We recall the definition.…”

Section: This Holds For Eachmentioning

confidence: 92%

“…The proof given in [12], which involves both algebraic and analytic ingredients, does not carry over easily to the nonarchimedean setting. One of our main results is the following adelic generalization of DeMarco's theorem:…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…We construct it by lifting ϕ to a polynomial map F : C 2 v → C 2 v , and using the homogeneous local height associated to the filled Julia set K F,v of this lift. This approach was inspired by DeMarco [12], who introduced the homogeneous capacity c 0 (K) for sets K ⊂ C 2 , and proved for archimedean v for each v. This is proved by relating the homogeneous transfinite diameter to the sectional capacity studied in [10] and [35]. The fact that v d 0 ∞ (K F,v ) = 1, which follows from the product formula applied to (0.1), is the key to the global lower bound mentioned above.…”

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confidence: 99%

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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: This Holds For Eachmentioning

confidence: 92%

Section: Annales De L'institut Fouriermentioning

confidence: 99%

mentioning

confidence: 99%

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Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Baker¹,

Rumely²

2006

Ann. inst. Fourier

101

154

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“…In 2000, De Marco [10,11] proved a formula relating the Lyapunov function L( f λ ) of a rational map to the critical dynamics of the family. It turns out that the canonical closed and positive (1,1)-current dd c λ L on the parameter space is exactly supported on the bifurcation locus (the observation that the bifurcation measure for the family z 2 + c is equal to the harmonic measure for the Mandelbrot set is due to N. Sibony, see [27]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

Bianchi

2018

Math. Ann.

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Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition for the volume of the iterates of the critical set. This generalizes to higher dimensions the well-known equivalence between stability and normality of the critical orbits in dimension one. We also introduce a notion of holomorphic motion of asymptotically all repelling cycles and prove its equivalence with other notions of stability. Our results allow us to generalize the theory of stability and bifurcation developed by Berteloot, Dupont and the author for the family of all endomorphisms of P k of a given degree to any arbitrary family of endomorphisms of P k or polynomial-like maps of large topological degree. Mathematics Subject Classification

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“…They also studied some properties of the bifurcation currents and obtained nice formulas for that currents in terms of the Green functions. We also refer to DeMarco, Dujardin-Favre, McMullen and Sibony [31,56,98,114] for results in dimension one. Notes.…”

Section: Thereforementioning

confidence: 99%

Dynamics in Several Complex Variables

Abate

2017

Lecture Notes in Mathematics

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The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.The first chapter deals with holomorphic endomorphisms of the projective space P k . We establish the first properties and give several constructions for the Green currents T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimal background. In order to help the reader, an appendix gives the basics on p.s.h. functions, positive closed currents and super-potentials on projective spaces. Some exercises are proposed and an extensive bibliography is given.

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Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity

Cited by 80 publications

References 19 publications

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

Dynamics in Several Complex Variables

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