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

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September 2011, 31(3): 985-996. doi: 10.3934/dcds.2011.31.985

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

1.	Toba National College of Maritime Technology, Mie 517-8501, Japan

Received June 2010 Revised February 2011 Published August 2011

Abstract
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We consider the dynamics of nondegenerate polynomial skew products on $\mathbb{C}^{2}$. The paper includes investigations of the existence of the Green and fiberwise Green functions of the maps, which induce generalized Green functions that are well-behaved on $\mathbb{C}^{2}$, and examples of the Green functions which are not defined on some curves in $\mathbb{C}^{2}$. Moreover, we consider the dynamics of the extensions of the maps to holomorphic or rational maps on weighted projective spaces.

Keywords: Complex dynamics, skew product, Green function..

Mathematics Subject Classification: Primary: 32H50; Secondary: 30D05, 37H9.

Citation: Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 985-996. doi: 10.3934/dcds.2011.31.985

References:

[1]	E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of $\mathbbC^k$, Amer. J. Math., 122 (2000), 153-212.
[2]	E. Bedford and J. Smillie, Polynomial diffeomorphisms of $\mathbbC^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.
[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.
[4]	T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières, (French) [Dynamics of semiregular polynomial maps], Ark. Mat., 42 (2004), 61-85.
[5]	T.-C. Dinh, R. Dujardin and N. Sibony, On the dynamics near infinity of some polynomial mappings in $\mathbbC^2$, Math. Ann., 333 (2005), 703-739. doi: 10.1007/s00208-005-0661-3.
[6]	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.
[7]	C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup., 40 (2007), 309-349.
[8]	C. Favre and M. Jonsson, Dynamical compactifications of $\mathbbC^2$, Ann. of Math., 173 (2011), 211-248. doi: 10.4007/annals.2011.173.1.6.
[9]	V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.
[10]	V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.
[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 $\mathbbC^2$, Kyushu J. Math., 52 (1998), 299-329. doi: 10.2206/kyushujm.52.299.
[13]	M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.
[14]	T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan, 46 (1994), 545-555. doi: 10.2969/jmsj/04630545.
[15]	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.
[16]	G. Vigny, Dynamics semi-conjugated to a subshift for some polynomial mappings in $\mathbbC^2$, Publ. Mat., 51 (2007), 201-222.

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.
[2]	E. Bedford and J. Smillie, Polynomial diffeomorphisms of $\mathbbC^2$: Currents, equilibrium measure and hyperbolicity, Invent. Math., 103 (1991), 69-99. doi: 10.1007/BF01239509.
[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.
[4]	T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières, (French) [Dynamics of semiregular polynomial maps], Ark. Mat., 42 (2004), 61-85.
[5]	T.-C. Dinh, R. Dujardin and N. Sibony, On the dynamics near infinity of some polynomial mappings in $\mathbbC^2$, Math. Ann., 333 (2005), 703-739. doi: 10.1007/s00208-005-0661-3.
[6]	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.
[7]	C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup., 40 (2007), 309-349.
[8]	C. Favre and M. Jonsson, Dynamical compactifications of $\mathbbC^2$, Ann. of Math., 173 (2011), 211-248. doi: 10.4007/annals.2011.173.1.6.
[9]	V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.
[10]	V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. doi: 10.1307/mmj/1100623417.
[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 $\mathbbC^2$, Kyushu J. Math., 52 (1998), 299-329. doi: 10.2206/kyushujm.52.299.
[13]	M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.
[14]	T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan, 46 (1994), 545-555. doi: 10.2969/jmsj/04630545.
[15]	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.
[16]	G. Vigny, Dynamics semi-conjugated to a subshift for some polynomial mappings in $\mathbbC^2$, Publ. Mat., 51 (2007), 201-222.

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