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

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

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.
Citation: Kohei Ueno. Weighted Green functions of polynomial skew products on $\mathbb{C}^2$. Discrete and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

K. Ueno, Polynomial skew products whose Julia sets have infinitely many symmetries, submitted.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[10]

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

[11]

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

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[19]

K. Ueno, Polynomial skew products whose Julia sets have infinitely many symmetries, submitted.

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