Geometry of currents, intersection theory and dynamics of horizontal-like maps

Dinh, Tien-Cuong; Sibony, Nessim

doi:10.5802/aif.2188

Cited by 34 publications

(31 citation statements)

References 27 publications

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“…These results have been applied in many questions of smooth dynamics. In the last several years more dynamical applications appeared in the study of the polynomial and rational dynamics in several complex variables, and in a more general context of the complexity of iterations [1,2,16,17,19,20,18,21,22,26,29,30,35,36,39,[43][44][45][46]51,52,64], as well as in the study of the behavior of discretized PDE's [42]. On the other hand, recently the C k -reparametrization theorem has been applied in the study of Anderson localization for Schrodinger operator on Z 2 with quasiperiodic potential [10,11].Yet another application appeared in counting rational points on and near algebraic varieties [47,54,49], see also [13,48].…”

Section: Discussionmentioning

confidence: 99%

“…In particular, this concerns the study of the semicontinuity modulus of the topological entropy in analytic families (see [57]), the problem of estimating the topological entropy in finite accuracy computations (see [45,44,1,2,57,64]) as well as the problem of bounding entropy of rational maps with singularities [17,19,20,18,29,30,35,36,51,43]. Also some number-theoretic questions posed in [47,49,53,54] lead, presumably, to the same kind of questions.…”

Section: Analytic Reparametrizationmentioning

confidence: 98%

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Analytic reparametrization of semi-algebraic sets

Yomdin

2008

Journal of Complexity

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In many problems in analysis, dynamics, and in their applications, it is important to subdivide objects under consideration into simple pieces, keeping control of high-order derivatives. It is known that semi-algebraic sets and mappings allow for such a controlled subdivision: this is the "C k reparametrization theorem" which is a high-order quantitative version of the well-known results on the existence of a triangulation of semialgebraic sets. In a C k -version we just require in addition that each simplex be represented as an image, under the "reparametrization mapping" , of the standard simplex, with all the derivatives of up to order k uniformly bounded. The main result of this paper is, that if we reparametrize all the set A but its small part of a size , we can do much more: not only to "kill" the derivatives, but also to bound uniformly the analytic complexity of the pieces, while their number remains of order log 1 . SummaryQuantitative information about geometric and analytic structure of algebraic and semi-algebraic sets is important in many problems of analysis, geometry, differential equations, dynamics, etc. In some applications it is enough to control just the rough topological information, like the number of simplices in the triangulation. In others the Lipschitzian or the C 1 bounds are important (see, for example, [40,41]). In some problems of analysis and dynamics the control of highorder derivatives is essential. It turns out that semi-algebraic sets and mappings allow for such a control.The main example is provided by the "C k reparametrization theorem" [55][56][57]34]. This can be considered as a high-order quantitative version of the well known result on the existence of a ଁ

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Section: Discussionmentioning

confidence: 99%

Section: Analytic Reparametrizationmentioning

confidence: 98%

Analytic reparametrization of semi-algebraic sets

Yomdin

2008

Journal of Complexity

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“…Given a positive closed (k, k) current R on Y × P m it follows from [Fed69] (se also [DS06b]) that the slices R y := R, π Y , y exist for a.e. y ∈ Y.…”

Section: Self-averagingmentioning

confidence: 99%

Zero distribution of random sparse polynomials

Bayraktar¹

2017

Michigan Math. J.

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Abstract. We study asymptotic zero distribution of random Laurent polynomials whose support are contained in dilates of a fixed integral polytope P as their degree grow. We consider a large class of probability distributions including the ones induced from i.i.d. random coefficients whose distribution law has bounded density with logarithmically decaying tails as well as moderate measures defined over the projectivized space of Laurent polynomials. We obtain a quantitative localized version of Bernstein-Kouchnirenko Theorem.

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“…Moreover, we do not need to restrict M to get the statement since Ω is compactly supported. This also follows from the compactness of horizontal positive closed currents with bounded slice mass, see [13].…”

Section: Definition 212mentioning

confidence: 94%

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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Geometry of currents, intersection theory and dynamics of horizontal-like maps

Cited by 34 publications

References 27 publications

Analytic reparametrization of semi-algebraic sets

Analytic reparametrization of semi-algebraic sets

Zero distribution of random sparse polynomials

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

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