2019, 15: 95-130. doi: 10.3934/jmd.2019014

Lattès maps and the interior of the bifurcation locus

LAMA, UMR8050, Université Paris-Est Marne-La-Vallée, 5 Boulevard Descartes, 77454 Champs-sur-Marne, France

Received  March 22, 2018 Revised  October 15, 2018 Published  May 2019

We study the phenomenon of robust bifurcations in the space of holomorphic maps of $ \mathbb{P}^2(\mathbb{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 $ \mathbb{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.

Citation: Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014
References:
[1]

I. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc., 39 (1964), 615-622.  doi: 10.1112/jlms/s1-39.1.615.

[2]

P. Berger, Generic family with robustly infinitely many sinks, Invent. Math., 205 (2016), 121-172.  doi: 10.1007/s00222-015-0632-6.

[3]

F. Berteloot and F. Bianchi, Perturbations d'exemples de Lattès et dimension de Hausdorff du lieu de bifurcation, J. Math. Pures Appl., 116 (2018), 161-Ű173. doi: 10.1016/j.matpur.2017.11.009.

[4]

F. BertelootF. Bianchi and C. Dupont, Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of $\mathbb{P}^{2}$, Ann. Sci. École Norm. Sup., 51 (2018), 215-262.  doi: 10.24033/asens.2355.

[5]

F. Berteloot and C. Dupont, Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80 (2005), 433-454.  doi: 10.4171/CMH/21.

[6]

S. Biebler, Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in $\mathbb{C}^{3}$, arXiv: 1611.02011v2, 2018.

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396.  doi: 10.2307/2118647.

[8]

G. T. Buzzard, Infinitely many periodic attractors for holomorphic maps of 2 variables, Ann. of Math., 145 (1997), 389-417.  doi: 10.2307/2951819.

[9]

M. Dabija and M. Jonsson, Algebraic webs invariant under endomorphisms, Publ. Math., 54 (2010), 137-148.  doi: 10.5565/PUBLMAT_54110_07.

[10]

R. Dujardin, Non-density of stability for holomorphic mappings on $\mathbb{P}^{k}$, J. Éc. polytech. Math., 4 (2017), 813-843.  doi: 10.5802/jep.57.

[11]

R. Dujardin and M. Lyubich, Stability and bifurcations for dissipative polynomial automorphisms of $\mathbb{C}^{2}$, Invent. Math., 200 (2015), 439-511.  doi: 10.1007/s00222-014-0535-y.

[12]

J. Kaneko and S. Tokugana, Complex crystallographic groups. Ⅱ, J. Math. Soc. Japan, 34 (1982), 595-605.  doi: 10.2969/jmsj/03440595.

[13]

M. Lyubich, An analysis of stability of the dynamics of rational functions, Teoriya Funk., Funk. Anal. Prilozh., 42 (1984), 72-81. 

[14]

R. MañéP. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup., 16 (1983), 193-217.  doi: 10.24033/asens.1446.

[15]

J. Milnor, On Lattès maps, in Dynamics on the Riemann Sphere, European Math. Soc., Zürich, 2006, 9–43. doi: 10.4171/011-1/1.

[16] S. MorosawaY. NishimuraM. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 66, Cambridge University Press, Cambridge, 2000. 
[17]

F. Rong, Lattès maps on $\mathbb{P}^{2}$, J. Math. Pures Appl., 93 (2010), 636-650.  doi: 10.1016/j.matpur.2009.10.002.

[18]

J. Taflin, Blenders near polynomial product maps of $\mathbb{C}^{2}$, arXiv: 1702.02115v2, 2017.

show all references

References:
[1]

I. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc., 39 (1964), 615-622.  doi: 10.1112/jlms/s1-39.1.615.

[2]

P. Berger, Generic family with robustly infinitely many sinks, Invent. Math., 205 (2016), 121-172.  doi: 10.1007/s00222-015-0632-6.

[3]

F. Berteloot and F. Bianchi, Perturbations d'exemples de Lattès et dimension de Hausdorff du lieu de bifurcation, J. Math. Pures Appl., 116 (2018), 161-Ű173. doi: 10.1016/j.matpur.2017.11.009.

[4]

F. BertelootF. Bianchi and C. Dupont, Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of $\mathbb{P}^{2}$, Ann. Sci. École Norm. Sup., 51 (2018), 215-262.  doi: 10.24033/asens.2355.

[5]

F. Berteloot and C. Dupont, Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80 (2005), 433-454.  doi: 10.4171/CMH/21.

[6]

S. Biebler, Persistent homoclinic tangencies and infinitely many sinks for residual sets of automorphisms of low degree in $\mathbb{C}^{3}$, arXiv: 1611.02011v2, 2018.

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396.  doi: 10.2307/2118647.

[8]

G. T. Buzzard, Infinitely many periodic attractors for holomorphic maps of 2 variables, Ann. of Math., 145 (1997), 389-417.  doi: 10.2307/2951819.

[9]

M. Dabija and M. Jonsson, Algebraic webs invariant under endomorphisms, Publ. Math., 54 (2010), 137-148.  doi: 10.5565/PUBLMAT_54110_07.

[10]

R. Dujardin, Non-density of stability for holomorphic mappings on $\mathbb{P}^{k}$, J. Éc. polytech. Math., 4 (2017), 813-843.  doi: 10.5802/jep.57.

[11]

R. Dujardin and M. Lyubich, Stability and bifurcations for dissipative polynomial automorphisms of $\mathbb{C}^{2}$, Invent. Math., 200 (2015), 439-511.  doi: 10.1007/s00222-014-0535-y.

[12]

J. Kaneko and S. Tokugana, Complex crystallographic groups. Ⅱ, J. Math. Soc. Japan, 34 (1982), 595-605.  doi: 10.2969/jmsj/03440595.

[13]

M. Lyubich, An analysis of stability of the dynamics of rational functions, Teoriya Funk., Funk. Anal. Prilozh., 42 (1984), 72-81. 

[14]

R. MañéP. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup., 16 (1983), 193-217.  doi: 10.24033/asens.1446.

[15]

J. Milnor, On Lattès maps, in Dynamics on the Riemann Sphere, European Math. Soc., Zürich, 2006, 9–43. doi: 10.4171/011-1/1.

[16] S. MorosawaY. NishimuraM. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 66, Cambridge University Press, Cambridge, 2000. 
[17]

F. Rong, Lattès maps on $\mathbb{P}^{2}$, J. Math. Pures Appl., 93 (2010), 636-650.  doi: 10.1016/j.matpur.2009.10.002.

[18]

J. Taflin, Blenders near polynomial product maps of $\mathbb{C}^{2}$, arXiv: 1702.02115v2, 2017.

Figure 1.  The yellow color stands for $\mathscr{U}_{x} \backslash (\mathscr{U}_{x} \cap \mathscr{U}''_{x})$, the red for $\mathscr{U}'_{x}$, the blue for $\mathscr{U}''_{x} \backslash \mathscr{U}'_{x}$. The arrows show a typical sequence of matrices: one multiplies $I_{2}$ by $I_{2}+M_{0}$ (with $M_{0} \in x \cdot V^{0} $) a finite number of times, then by $I_{2}+M_{p}$ (with $M_{p} \in x \cdot V^{p} $)
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