2022, 18: 69-99. doi: 10.3934/jmd.2022003

Higher bifurcations for polynomial skew products

1. 

Université d'Orléans, Institut Denis Poisson, UMR CNRS 7013, 45067 Orléans Cedex 2, France

2. 

CNRS, Univ. Lille, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France

Received  June 26, 2020 Revised  November 18, 2021 Published  March 2022

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where $ k $ critical points bifurcate independently, with $ k $ up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.

Citation: Matthieu Astorg, Fabrizio Bianchi. Higher bifurcations for polynomial skew products. Journal of Modern Dynamics, 2022, 18: 69-99. doi: 10.3934/jmd.2022003
References:
[1]

M. Astorg and F. Bianchi, Hyperbolicity and bifurcations in holomorphic families of polynomial skew products, preprint, arXiv: 1801.01460, 2018.

[2]

M. AstorgX. BuffR. DujardinH. Peters and J. Raissy, A two-dimensional polynomial mapping with a wandering Fatou component, Ann. of Math., 184 (2016), 263-313.  doi: 10.4007/annals.2016.184.1.2.

[3]

M. AstorgT. GauthierN. Mihalache and G. Vigny, Collet, Eckmann and the bifurcation measure, Invent. Math., 217 (2019), 749-797.  doi: 10.1007/s00222-019-00874-5.

[4]

G. Bassanelli and F. Berteloot, Bifurcation currents in holomorphic dynamics on $\mathbb {P}^k$, J. Reine Angew. Math., 608 (2007), 201-235.  doi: 10.1515/CRELLE.2007.058.

[5]

F. Berteloot and F. Bianchi, Perturbations d'exemples de Lattès et dimension de {H}ausdorff du lieu de bifurcation, J. Math. Pures Appl. (9), 116 (2018), 161-173.  doi: 10.1016/j.matpur.2017.11.009.

[6]

F. BertelootF. Bianchi and C. Dupont, Dynamical stability and {L}yapunov exponents for holomorphic endomorphisms of $\mathbb{P}^k$, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 215-262.  doi: 10.24033/asens.2355.

[7]

F. Berteloot and C. Dupont, A distortion theorem for iterated inverse branches of holomorphic endomorphisms of $\mathbb{P}^k$, J. Lond. Math. Soc. (2), 99 (2019), 153-172.  doi: 10.1112/jlms.12163.

[8]

F. BertelootC. Dupont and L. Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble), 58 (2008), 2137-2168.  doi: 10.5802/aif.2409.

[9]

F. Bianchi and T.-C. Dinh, Existence and properties of equilibrium states of holomorphic endomorphisms of $\mathbb {P}^k$, preprint, arXiv: 2007.04595, 2020.

[10]

F. Bianchi, Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree, Math. Ann., 373 (2019), 901-928.  doi: 10.1007/s00208-018-1642-7.

[11]

F. Bianchi and J. Taflin, Bifurcations in the elementary {D}esboves family, Proc. Amer. Math. Soc., 145 (2017), 4337-4343.  doi: 10.1090/proc/13579.

[12]

S. Biebler, Lattès maps and the interior of the bifurcation locus, J. Mod. Dyn., 15 (2019), 95-130.  doi: 10.3934/jmd.2019014.

[13]

J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de $\mathbb {C} \mathbb {P}^k$, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.

[14]

X. Buff and A. Epstein, Bifurcation measure and postcritically finite rational maps, in Complex Dynamics: Families and Friends, A K Peters, Wellesley, MA, 2009,491–512. doi: 10.1201/b10617-16.

[15]

I. P. Cornfeld, S. V. Fomin, and Y. G. Sina$\mathop {\rm{i}}\limits^ \vee $, Ergodic Theory, Fundamental Principles of Mathematical Sciences, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[16]

L. DeMarco, Dynamics of rational maps: A current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.

[17]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, Lecture Notes in Math., 1998, Springer, Berlin, 2010,165–294. doi: 10.1007/978-3-642-13171-4_4.

[18]

R. Dujardin and C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032.  doi: 10.1353/ajm.0.0009.

[19]

R. Dujardin, Bifurcation currents and equidistribution in parameter space, Frontiers in Complex Dynamics, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, 2014,515–566.

[20]

R. Dujardin, A non-laminar dynamical green current, Math. Ann., 365 (2016), 77-91.  doi: 10.1007/s00208-015-1274-0.

[21]

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.

[22]

C. Dupont and J. Taflin, Dynamics of fibered endomorphisms of $\mathbb{P}^k$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22 (2021), 53-78.  doi: 10.2422/2036-2145.201811_017.

[23]

T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér.(4), 45 (2012), 947–984. doi: 10.24033/asens.2181.

[24]

T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: Multipliers and degree $d$ polynomials, Math. Ann., 366 (2016), 1-30.  doi: 10.1007/s00208-015-1297-6.

[25]

I. Gorbovickis, Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable, Ergodic Theory Dynam. Systems, 36 (2016), 1156-1166.  doi: 10.1017/etds.2014.103.

[26]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Fundamental Principles of Mathematical Sciences, 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.

[27]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.  doi: 10.5169/seals-66687.

[28]

M. Jonsson, Dynamics of polynomial skew products on $\mathbb{C}^2$, Math. Ann., 314 (1999), 403-447.  doi: 10.1007/s002080050301.

[29]

C. T. McMullen, The Mandelbrot set is universal, in The Mandelbrot Set, Themes and Variations, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, Cambridge, 2000, 1–17.

[30]

W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[31]

N.-M. Pham, Lyapunov exponents and bifurcation current for polynomial-like maps, preprint, arXiv: math/0512557, 2005.

[32]

F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math., 80 (1985), 161-179.  doi: 10.1007/BF01388554.

[33]

F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, in Proceedings of International Congress of Mathematicians, World Sci. Publ., Hackensack, NJ, 2018, 2087–2112.

[34]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.

[35]

F. Przytycki and A. Zdunik, On Hausdorff dimension of polynomial not totally disconnected Julia sets, Bull. Lond. Math. Soc., 53 (2021), 1674-1691.  doi: 10.1112/blms.12519.

[36]

J. Taflin, Blenders near polynomial product maps of $\mathbb{C}^2$, J. Eur. Math. Soc., 23 (2021), 3555-3589.  doi: 10.4171/JEMS/1076.

[37]

M. Urbański and A. Zdunik, Equilibrium measures for holomorphic endomorphisms of complex projective spaces, Fund. Math., 220 (2013), 23-69.  doi: 10.4064/fm220-1-3.

[38]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math., 99 (1990), 627-649.  doi: 10.1007/BF01234434.

show all references

References:
[1]

M. Astorg and F. Bianchi, Hyperbolicity and bifurcations in holomorphic families of polynomial skew products, preprint, arXiv: 1801.01460, 2018.

[2]

M. AstorgX. BuffR. DujardinH. Peters and J. Raissy, A two-dimensional polynomial mapping with a wandering Fatou component, Ann. of Math., 184 (2016), 263-313.  doi: 10.4007/annals.2016.184.1.2.

[3]

M. AstorgT. GauthierN. Mihalache and G. Vigny, Collet, Eckmann and the bifurcation measure, Invent. Math., 217 (2019), 749-797.  doi: 10.1007/s00222-019-00874-5.

[4]

G. Bassanelli and F. Berteloot, Bifurcation currents in holomorphic dynamics on $\mathbb {P}^k$, J. Reine Angew. Math., 608 (2007), 201-235.  doi: 10.1515/CRELLE.2007.058.

[5]

F. Berteloot and F. Bianchi, Perturbations d'exemples de Lattès et dimension de {H}ausdorff du lieu de bifurcation, J. Math. Pures Appl. (9), 116 (2018), 161-173.  doi: 10.1016/j.matpur.2017.11.009.

[6]

F. BertelootF. Bianchi and C. Dupont, Dynamical stability and {L}yapunov exponents for holomorphic endomorphisms of $\mathbb{P}^k$, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 215-262.  doi: 10.24033/asens.2355.

[7]

F. Berteloot and C. Dupont, A distortion theorem for iterated inverse branches of holomorphic endomorphisms of $\mathbb{P}^k$, J. Lond. Math. Soc. (2), 99 (2019), 153-172.  doi: 10.1112/jlms.12163.

[8]

F. BertelootC. Dupont and L. Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble), 58 (2008), 2137-2168.  doi: 10.5802/aif.2409.

[9]

F. Bianchi and T.-C. Dinh, Existence and properties of equilibrium states of holomorphic endomorphisms of $\mathbb {P}^k$, preprint, arXiv: 2007.04595, 2020.

[10]

F. Bianchi, Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree, Math. Ann., 373 (2019), 901-928.  doi: 10.1007/s00208-018-1642-7.

[11]

F. Bianchi and J. Taflin, Bifurcations in the elementary {D}esboves family, Proc. Amer. Math. Soc., 145 (2017), 4337-4343.  doi: 10.1090/proc/13579.

[12]

S. Biebler, Lattès maps and the interior of the bifurcation locus, J. Mod. Dyn., 15 (2019), 95-130.  doi: 10.3934/jmd.2019014.

[13]

J.-Y. Briend and J. Duval, Exposants de Liapounoff et distribution des points périodiques d'un endomorphisme de $\mathbb {C} \mathbb {P}^k$, Acta Math., 182 (1999), 143-157.  doi: 10.1007/BF02392572.

[14]

X. Buff and A. Epstein, Bifurcation measure and postcritically finite rational maps, in Complex Dynamics: Families and Friends, A K Peters, Wellesley, MA, 2009,491–512. doi: 10.1201/b10617-16.

[15]

I. P. Cornfeld, S. V. Fomin, and Y. G. Sina$\mathop {\rm{i}}\limits^ \vee $, Ergodic Theory, Fundamental Principles of Mathematical Sciences, 245, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[16]

L. DeMarco, Dynamics of rational maps: A current on the bifurcation locus, Math. Res. Lett., 8 (2001), 57-66.  doi: 10.4310/MRL.2001.v8.n1.a7.

[17]

T.-C. Dinh and N. Sibony, Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, Lecture Notes in Math., 1998, Springer, Berlin, 2010,165–294. doi: 10.1007/978-3-642-13171-4_4.

[18]

R. Dujardin and C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math., 130 (2008), 979-1032.  doi: 10.1353/ajm.0.0009.

[19]

R. Dujardin, Bifurcation currents and equidistribution in parameter space, Frontiers in Complex Dynamics, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, 2014,515–566.

[20]

R. Dujardin, A non-laminar dynamical green current, Math. Ann., 365 (2016), 77-91.  doi: 10.1007/s00208-015-1274-0.

[21]

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.

[22]

C. Dupont and J. Taflin, Dynamics of fibered endomorphisms of $\mathbb{P}^k$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22 (2021), 53-78.  doi: 10.2422/2036-2145.201811_017.

[23]

T. Gauthier, Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér.(4), 45 (2012), 947–984. doi: 10.24033/asens.2181.

[24]

T. Gauthier, Equidistribution towards the bifurcation current Ⅰ: Multipliers and degree $d$ polynomials, Math. Ann., 366 (2016), 1-30.  doi: 10.1007/s00208-015-1297-6.

[25]

I. Gorbovickis, Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable, Ergodic Theory Dynam. Systems, 36 (2016), 1156-1166.  doi: 10.1017/etds.2014.103.

[26]

H. Grauert and R. Remmert, Coherent Analytic Sheaves, Fundamental Principles of Mathematical Sciences, 265, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69582-7.

[27]

M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.  doi: 10.5169/seals-66687.

[28]

M. Jonsson, Dynamics of polynomial skew products on $\mathbb{C}^2$, Math. Ann., 314 (1999), 403-447.  doi: 10.1007/s002080050301.

[29]

C. T. McMullen, The Mandelbrot set is universal, in The Mandelbrot Set, Themes and Variations, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, Cambridge, 2000, 1–17.

[30]

W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[31]

N.-M. Pham, Lyapunov exponents and bifurcation current for polynomial-like maps, preprint, arXiv: math/0512557, 2005.

[32]

F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math., 80 (1985), 161-179.  doi: 10.1007/BF01388554.

[33]

F. Przytycki, Thermodynamic formalism methods in one-dimensional real and complex dynamics, in Proceedings of International Congress of Mathematicians, World Sci. Publ., Hackensack, NJ, 2018, 2087–2112.

[34]

F. Przytycki and M. Urbański, Conformal Fractals: Ergodic Theory Methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, 2010. doi: 10.1017/CBO9781139193184.

[35]

F. Przytycki and A. Zdunik, On Hausdorff dimension of polynomial not totally disconnected Julia sets, Bull. Lond. Math. Soc., 53 (2021), 1674-1691.  doi: 10.1112/blms.12519.

[36]

J. Taflin, Blenders near polynomial product maps of $\mathbb{C}^2$, J. Eur. Math. Soc., 23 (2021), 3555-3589.  doi: 10.4171/JEMS/1076.

[37]

M. Urbański and A. Zdunik, Equilibrium measures for holomorphic endomorphisms of complex projective spaces, Fund. Math., 220 (2013), 23-69.  doi: 10.4064/fm220-1-3.

[38]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math., 99 (1990), 627-649.  doi: 10.1007/BF01234434.

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