October  2020, 40(12): 6611-6633. doi: 10.3934/dcds.2020262

The Mandelbrot set is the shadow of a Julia set

1. 

Université Toulouse 3, Institut Mathématique de Toulouse, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France

2. 

Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076

* Corresponding author: Tien-Cuong Dinh

Received  September 2019 Revised  March 2020 Published  July 2020

Fund Project: The second author was supported by the NUS grants C-146-000-047-001 and R-146-000-248-114

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the set of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.

Citation: François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
References:
[1]

T. Ahn, Equidistribution in higher codimension for holomorphic endomorphisms of $ \mathbb{P}^k$, Trans. Amer. Math. Soc., 368 (2016), no. 5, 3359–3388. doi: 10.1090/tran/6539.  Google Scholar

[2]

F. Berteloot, Bifurcation currents in holomorphic families of rational maps, In Pluripotential Theory, Lect. Notes in Math., Vol. 2075, CIME Fundation subseries, Springer, Heidelberg, 2013, 1–93. doi: 10.1007/978-3-642-36421-1_1.  Google Scholar

[3]

F. Berteloot and V. Mayer, Rudiments de dynamique holomorphe, Société Mathématique de France, Paris, EDP Sciences, Les Ulis, 2001.  Google Scholar

[4]

F. Berteloot and F. Bianchi, Stability in projective holomorphic dynamics, Polish Acad. Sci. Inst. Math., Vol. 115, Warsaw, 2018, 1–35.  Google Scholar

[5]

X. Buff and T. Gauthier, Quadratic polynomials, multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), no. 7, 3011–3017. doi: 10.1090/S0002-9939-2015-12506-3.  Google Scholar

[6]

L. Carleson and T. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[7]

L. DeMarco, Dynamics of rational maps : Lyapunov exponents, bifurcations, and capacity, Math. Ann., 326 (2003), 43-73.  doi: 10.1007/s00208-002-0404-7.  Google Scholar

[8]

T.-C. Dinh and N. Sibony, Une borne supérieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005), 1637-1644.  doi: 10.4007/annals.2005.161.1637.  Google Scholar

[9]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier, 56 (2006), no. 2,423–457. doi: 10.5802/aif.2188.  Google Scholar

[10]

T.-C. Dinh and N. Sibony, Pull-back of currents by holomorphic maps, Manuscripta Math., 123 (2007), no. 3,357–371. doi: 10.1007/s00229-007-0103-5.  Google Scholar

[11]

T.-C. Dinh and N. Sibony, Equidistribution speed for endomorphisms of projective spaces, Math. Ann., 347 (2010), no. 3,613–626. doi: 10.1007/s00208-009-0445-2.  Google Scholar

[12]

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

[13]

T.-C. Dinh and N. Sibony, Density of positive closed currents, a theory of non-generic intersections, J. Algebraic Geom., 27 (2018), 497-551.  doi: 10.1090/jag/711.  Google Scholar

[14]

R. Dujardin, Bifurcation currents and equidistribution on parameter space, Frontiers of Complex Dynamics, Princeton University Press, Princeton, NJ, 2014,515-566.  Google Scholar

[15]

C. Favre and M. Jonsson, Brolin's theorem for curves in two complex dimensions, Ann. Inst. Fourier 53 (2003), no. 5, 1461–1501.  Google Scholar

[16]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 439, Complex potential theory (Montreal, PQ, 1993), Kluwer Acad. Publ., Dordrecht, 1994,131–186.  Google Scholar

[17]

M. Y. Lyubich, Some typical properties of the dynamics of rational mappings, Uspekhi Mat. Nauk, 38 (1983), 154-155.   Google Scholar

[18]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps., Ann. Sci. École Norm. Sup. (4), 16 (1983), 193–217.  Google Scholar

[19]

N. Sibony, Exposés à Orsay non publiés, Course UCLA, (1981). Google Scholar

[20]

N. Sibony, Dynamique des applications rationnelles de $\mathbb{P}^k$., Panor. Synthèses, 8 (1999), 97–185.  Google Scholar

[21]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécialisés, Vol. 10, Société Mathématique de France, Paris, 2002. doi: 10.1017/CBO9780511615344.  Google Scholar

[22]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbb{P}^k$, Adv. Math., 227 (2011), no. 5, 2059–2081. doi: 10.1016/j.aim.2011.04.010.  Google Scholar

[23]

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975,590 pp.  Google Scholar

show all references

References:
[1]

T. Ahn, Equidistribution in higher codimension for holomorphic endomorphisms of $ \mathbb{P}^k$, Trans. Amer. Math. Soc., 368 (2016), no. 5, 3359–3388. doi: 10.1090/tran/6539.  Google Scholar

[2]

F. Berteloot, Bifurcation currents in holomorphic families of rational maps, In Pluripotential Theory, Lect. Notes in Math., Vol. 2075, CIME Fundation subseries, Springer, Heidelberg, 2013, 1–93. doi: 10.1007/978-3-642-36421-1_1.  Google Scholar

[3]

F. Berteloot and V. Mayer, Rudiments de dynamique holomorphe, Société Mathématique de France, Paris, EDP Sciences, Les Ulis, 2001.  Google Scholar

[4]

F. Berteloot and F. Bianchi, Stability in projective holomorphic dynamics, Polish Acad. Sci. Inst. Math., Vol. 115, Warsaw, 2018, 1–35.  Google Scholar

[5]

X. Buff and T. Gauthier, Quadratic polynomials, multipliers and equidistribution, Proc. Amer. Math. Soc., 143 (2015), no. 7, 3011–3017. doi: 10.1090/S0002-9939-2015-12506-3.  Google Scholar

[6]

L. Carleson and T. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9.  Google Scholar

[7]

L. DeMarco, Dynamics of rational maps : Lyapunov exponents, bifurcations, and capacity, Math. Ann., 326 (2003), 43-73.  doi: 10.1007/s00208-002-0404-7.  Google Scholar

[8]

T.-C. Dinh and N. Sibony, Une borne supérieure de l'entropie topologique d'une application rationnelle, Ann. of Math., 161 (2005), 1637-1644.  doi: 10.4007/annals.2005.161.1637.  Google Scholar

[9]

T.-C. Dinh and N. Sibony, Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier, 56 (2006), no. 2,423–457. doi: 10.5802/aif.2188.  Google Scholar

[10]

T.-C. Dinh and N. Sibony, Pull-back of currents by holomorphic maps, Manuscripta Math., 123 (2007), no. 3,357–371. doi: 10.1007/s00229-007-0103-5.  Google Scholar

[11]

T.-C. Dinh and N. Sibony, Equidistribution speed for endomorphisms of projective spaces, Math. Ann., 347 (2010), no. 3,613–626. doi: 10.1007/s00208-009-0445-2.  Google Scholar

[12]

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

[13]

T.-C. Dinh and N. Sibony, Density of positive closed currents, a theory of non-generic intersections, J. Algebraic Geom., 27 (2018), 497-551.  doi: 10.1090/jag/711.  Google Scholar

[14]

R. Dujardin, Bifurcation currents and equidistribution on parameter space, Frontiers of Complex Dynamics, Princeton University Press, Princeton, NJ, 2014,515-566.  Google Scholar

[15]

C. Favre and M. Jonsson, Brolin's theorem for curves in two complex dimensions, Ann. Inst. Fourier 53 (2003), no. 5, 1461–1501.  Google Scholar

[16]

J. E. Fornæss and N. Sibony, Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 439, Complex potential theory (Montreal, PQ, 1993), Kluwer Acad. Publ., Dordrecht, 1994,131–186.  Google Scholar

[17]

M. Y. Lyubich, Some typical properties of the dynamics of rational mappings, Uspekhi Mat. Nauk, 38 (1983), 154-155.   Google Scholar

[18]

R. Mañé, P. Sad and D. Sullivan, On the dynamics of rational maps., Ann. Sci. École Norm. Sup. (4), 16 (1983), 193–217.  Google Scholar

[19]

N. Sibony, Exposés à Orsay non publiés, Course UCLA, (1981). Google Scholar

[20]

N. Sibony, Dynamique des applications rationnelles de $\mathbb{P}^k$., Panor. Synthèses, 8 (1999), 97–185.  Google Scholar

[21]

C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécialisés, Vol. 10, Société Mathématique de France, Paris, 2002. doi: 10.1017/CBO9780511615344.  Google Scholar

[22]

J. Taflin, Equidistribution speed towards the Green current for endomorphisms of $ \mathbb{P}^k$, Adv. Math., 227 (2011), no. 5, 2059–2081. doi: 10.1016/j.aim.2011.04.010.  Google Scholar

[23]

M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publishing Co., New York, 1975,590 pp.  Google Scholar

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