Algebraic webs invariant under endomorphisms

Dabija, Marius; Jönsson, Mattias

doi:10.5565/publmat_54110_07

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…One of the maps L k in {L, L 2 , L 3 , L 6 } is preserving an algebraic web associated to a smooth cubic (see [8] for this notion). This implies (see the remark after Theorem A in [8]) that the critical set of L k is sent after one iteration into the set of critical values of Π which is a curve PC. In this second case, we have that L k (PC) ⊂ PC and L k induces by restriction a map on PC.…”

Section: A Periodic Orbit In the Postcritical Setmentioning

confidence: 99%

Lattès maps and the interior of the bifurcation locus

Biebler

2019

Journal of Modern Dynamics

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We study the phenomenon of robust bifurcations in the space of holomorphic maps of P 2 (C). We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C 2 with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry.

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Section: A Periodic Orbit In the Postcritical Setmentioning

confidence: 99%

Lattès maps and the interior of the bifurcation locus

Biebler

2019

Journal of Modern Dynamics

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“…Indeed, it is easy to check, using the classifications in [DJ10] and [FP15] and the proof of Theorem 1.1, that this is the case when k = 2 and the family (L a ) a∈X defines a web with algebraic leaves. However, in some of these examples S = T f .…”

Section: Introductionmentioning

confidence: 96%

Dynamics of fibered endomorphisms of CP(k)

Dupont¹,

Taflin²

2021

Annali Scuola Normale Superiore - Classe Di Scienze

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We study the structure and the Lyapunov exponents of the equilibrium measure of endomorphisms of P k preserving a fibration. We extend the decomposition of the equilibrium measure obtained by Jonsson for polynomial skew products of C 2 . We also show that the sum of the sectional exponents satisfies a Bedford-Jonsson formula when the fibration is linear, and that this function is plurisubharmonic on families of fibered endomorphisms. In particular, the sectional part of the bifurcation current is a closed positive current on the parameter space.

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“…Many authors have studied meromorphic (and rational) maps that preserve foliations and algebraic webs, including [21,8,31,11,12,7,13]. Since the proof of Theorem 1.1 is self-contained and provides insight on the mechanism that prevents ϕ from preserving a foliation, we will provide a direct proof rather than appealing to results from these previous papers.…”

Section: Introductionmentioning

confidence: 99%