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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 $\mathbb{C}^2$

Kohei Ueno 1,

1. 

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

Received  June 2010 Revised  February 2011 Published  August 2011

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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $\mathbbC^2$: Currents, equilibrium measure and hyperbolicity,, Invent. Math., 103 (1991), 69.  doi: 10.1007/BF01239509.  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.  doi: 10.1017/S0143385708000047.  Google Scholar

[4]

T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières,, (French) [Dynamics of semiregular polynomial maps], 42 (2004), 61.   Google Scholar

[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.  doi: 10.1007/s00208-005-0661-3.  Google Scholar

[6]

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

[7]

C. Favre and M. Jonsson, Eigenvaluations,, Ann. Sci. École Norm. Sup., 40 (2007), 309.   Google Scholar

[8]

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

[9]

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

[10]

V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$,, Michigan Math. J., 52 (2004), 627.  doi: 10.1307/mmj/1100623417.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

T. Ueda, Fatou sets in complex dynamics on projective spaces,, J. Math. Soc. Japan, 46 (1994), 545.  doi: 10.2969/jmsj/04630545.  Google Scholar

[15]

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

[16]

G. Vigny, Dynamics semi-conjugated to a subshift for some polynomial mappings in $\mathbbC^2$,, Publ. Mat., 51 (2007), 201.   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.   Google Scholar

[2]

E. Bedford and J. Smillie, Polynomial diffeomorphisms of $\mathbbC^2$: Currents, equilibrium measure and hyperbolicity,, Invent. Math., 103 (1991), 69.  doi: 10.1007/BF01239509.  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.  doi: 10.1017/S0143385708000047.  Google Scholar

[4]

T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières,, (French) [Dynamics of semiregular polynomial maps], 42 (2004), 61.   Google Scholar

[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.  doi: 10.1007/s00208-005-0661-3.  Google Scholar

[6]

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

[7]

C. Favre and M. Jonsson, Eigenvaluations,, Ann. Sci. École Norm. Sup., 40 (2007), 309.   Google Scholar

[8]

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

[9]

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

[10]

V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$,, Michigan Math. J., 52 (2004), 627.  doi: 10.1307/mmj/1100623417.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

T. Ueda, Fatou sets in complex dynamics on projective spaces,, J. Math. Soc. Japan, 46 (1994), 545.  doi: 10.2969/jmsj/04630545.  Google Scholar

[15]

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

[16]

G. Vigny, Dynamics semi-conjugated to a subshift for some polynomial mappings in $\mathbbC^2$,, Publ. Mat., 51 (2007), 201.   Google Scholar

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