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May  2014, 34(5): 2283-2305. doi: 10.3934/dcds.2014.34.2283

Weighted Green functions of polynomial skew products on $\mathbb{C}^2$

Kohei Ueno 1,

1. 

Toba National College of Maritime Technology, Mie 517-8501

Received  December 2012 Revised  August 2013 Published  October 2013

We study the dynamics of polynomial skew products on $\mathbb{C}^2$. By using suitable weights, we prove the existence of several types of Green functions. Largely, continuity and plurisubharmonicity follow. Moreover, it relates to the dynamics of the rational extensions to weighted projective spaces.
Keywords: Green function., skew product, Complex dynamics.
Mathematics Subject Classification: Primary: 32H50; Secondary: 30D0.
Citation: Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283
References:
[1]

E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of $\mathbbC^k$, Amer. J. Math., 122 (2000), 153-212. doi: 10.1353/ajm.2000.0001.  Google Scholar

[2]

S. Boucksom, C. Favre and M. Jonsson, Degree growth of meromorphic surface maps, Duke Math. J., 141 (2008), 519-538. doi: 10.1215/00127094-2007-004.  Google Scholar

[3]

L. DeMarco and S. L. Hruska, Axiom A polynomial skew products of $\mathbbC^2$ and their postcritical sets, Ergodic Theory Dynam. Systems, 28 (2008), 1749-1779. doi: 10.1017/S0143385708000047.  Google Scholar

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J. Diller and V. Guedj, Regularity of dynamical Green's functions, Trans. Amer. Math. Soc., 361 (2009), 4783-4805. doi: 10.1090/S0002-9947-09-04740-0.  Google Scholar

[5]

C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, (French) [Dynamics of rational mappings of multiprojective spaces], Indiana Univ. Math. J., 50 (2001), 881-934. doi: 10.1512/iumj.2001.50.1880.  Google Scholar

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C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853. Springer-Verlag, Berlin, 2004. doi: 10.1007/b100262.  Google Scholar

[7]

C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 309-349. doi: 10.1016/j.ansens.2007.01.002.  Google Scholar

[8]

C. Favre and M. Jonsson, Dynamical compactifications of $\mathbbC^2$, Ann. of Math. (2), 173 (2011), 211-249. doi: 10.4007/annals.2011.173.1.6.  Google Scholar

[9]

J. E. Fornæss and N. Sibony, Complex Dynamics in Higher Dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), 135-182, Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995.  Google Scholar

[10]

V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.  Google Scholar

[11]

S.-M. Heinemann, Julia sets for holomorphic endomorphisms of $\mathbbC^n$, Ergodic Theory Dynam. Systems, 16 (1996), 1275-1296. doi: 10.1017/S0143385700010026.  Google Scholar

[12]

S.-M. Heinemann, Julia sets of skew products in $\mathbbC^2$, Kyushu J. Math., 52 (1998), 299-329. doi: 10.2206/kyushujm.52.299.  Google Scholar

[13]

S. L. Hruska and R. K. W. Roeder, Topology of Fatou components for endomorphisms of $\mathbb{CP}^k$: linking with the Green's current, Fund. Math., 210 (2010), 73-98. doi: 10.4064/fm210-1-4.  Google Scholar

[14]

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

[15]

R. K. W. Roeder, A dichotomy for Fatou components of polynomial skew products, Conform. Geom. Dyn., 15 (2011), 7-19. doi: 10.1090/S1088-4173-2011-00223-2.  Google Scholar

[16]

K. Ueno, Symmetries of Julia sets of nondegenerate polynomial skew products on $\mathbbC^2$, Michigan Math. J., 59 (2010), 153-168. doi: 10.1307/mmj/1272376030.  Google Scholar

[17]

K. Ueno, Weighted Green functions of nondegenerate polynomial skew products on $\mathbbC^2$, Discrete Contin. Dyn. Syst., 31 (2011), 985-996. doi: 10.3934/dcds.2011.31.985.  Google Scholar

[18]

K. Ueno, Fiberwise Green functions of skew products semiconjugate to some polynomial products on $\mathbbC^2$, Kodai Math. J., 35 (2012), 345-357. doi: 10.2996/kmj/1341401055.  Google Scholar

[19]

K. Ueno, Polynomial skew products whose Julia sets have infinitely many symmetries,, submitted., ().   Google Scholar

show all references

References:
[1]

E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of $\mathbbC^k$, Amer. J. Math., 122 (2000), 153-212. doi: 10.1353/ajm.2000.0001.  Google Scholar

[2]

S. Boucksom, C. Favre and M. Jonsson, Degree growth of meromorphic surface maps, Duke Math. J., 141 (2008), 519-538. doi: 10.1215/00127094-2007-004.  Google Scholar

[3]

L. DeMarco and S. L. Hruska, Axiom A polynomial skew products of $\mathbbC^2$ and their postcritical sets, Ergodic Theory Dynam. Systems, 28 (2008), 1749-1779. doi: 10.1017/S0143385708000047.  Google Scholar

[4]

J. Diller and V. Guedj, Regularity of dynamical Green's functions, Trans. Amer. Math. Soc., 361 (2009), 4783-4805. doi: 10.1090/S0002-9947-09-04740-0.  Google Scholar

[5]

C. Favre and V. Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, (French) [Dynamics of rational mappings of multiprojective spaces], Indiana Univ. Math. J., 50 (2001), 881-934. doi: 10.1512/iumj.2001.50.1880.  Google Scholar

[6]

C. Favre and M. Jonsson, The Valuative Tree, Lecture Notes in Mathematics, 1853. Springer-Verlag, Berlin, 2004. doi: 10.1007/b100262.  Google Scholar

[7]

C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 309-349. doi: 10.1016/j.ansens.2007.01.002.  Google Scholar

[8]

C. Favre and M. Jonsson, Dynamical compactifications of $\mathbbC^2$, Ann. of Math. (2), 173 (2011), 211-249. doi: 10.4007/annals.2011.173.1.6.  Google Scholar

[9]

J. E. Fornæss and N. Sibony, Complex Dynamics in Higher Dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), 135-182, Ann. of Math. Stud., 137, Princeton Univ. Press, Princeton, NJ, 1995.  Google Scholar

[10]

V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.  Google Scholar

[11]

S.-M. Heinemann, Julia sets for holomorphic endomorphisms of $\mathbbC^n$, Ergodic Theory Dynam. Systems, 16 (1996), 1275-1296. doi: 10.1017/S0143385700010026.  Google Scholar

[12]

S.-M. Heinemann, Julia sets of skew products in $\mathbbC^2$, Kyushu J. Math., 52 (1998), 299-329. doi: 10.2206/kyushujm.52.299.  Google Scholar

[13]

S. L. Hruska and R. K. W. Roeder, Topology of Fatou components for endomorphisms of $\mathbb{CP}^k$: linking with the Green's current, Fund. Math., 210 (2010), 73-98. doi: 10.4064/fm210-1-4.  Google Scholar

[14]

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

[15]

R. K. W. Roeder, A dichotomy for Fatou components of polynomial skew products, Conform. Geom. Dyn., 15 (2011), 7-19. doi: 10.1090/S1088-4173-2011-00223-2.  Google Scholar

[16]

K. Ueno, Symmetries of Julia sets of nondegenerate polynomial skew products on $\mathbbC^2$, Michigan Math. J., 59 (2010), 153-168. doi: 10.1307/mmj/1272376030.  Google Scholar

[17]

K. Ueno, Weighted Green functions of nondegenerate polynomial skew products on $\mathbbC^2$, Discrete Contin. Dyn. Syst., 31 (2011), 985-996. doi: 10.3934/dcds.2011.31.985.  Google Scholar

[18]

K. Ueno, Fiberwise Green functions of skew products semiconjugate to some polynomial products on $\mathbbC^2$, Kodai Math. J., 35 (2012), 345-357. doi: 10.2996/kmj/1341401055.  Google Scholar

[19]

K. Ueno, Polynomial skew products whose Julia sets have infinitely many symmetries,, submitted., ().   Google Scholar

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