# American Institute of Mathematical Sciences

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

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.
Citation: Kohei Ueno. Weighted Green functions of nondegenerate polynomial skew products on $\mathbb{C}^2$. Discrete & 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.  Google Scholar [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.  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] T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières, (French) [Dynamics of semiregular polynomial maps], Ark. Mat., 42 (2004), 61-85.  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), 703-739. 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], Indiana Univ. Math. J., 50 (2001), 881-934. doi: 10.1512/iumj.2001.50.1880.  Google Scholar [7] C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup., 40 (2007), 309-349.  Google Scholar [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.  Google Scholar [9] V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.  Google Scholar [10] V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. 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), 1275-1296. 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] M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.  Google Scholar [14] T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan, 46 (1994), 545-555. 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), 153-168. 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), 201-222.  Google Scholar

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##### References:
 [1] E. Bedford and M. Jonsson, Dynamics of regular polynomial endomorphisms of $\mathbbC^k$, Amer. J. Math., 122 (2000), 153-212.  Google Scholar [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.  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] T.-C. Dinh and N. Sibony, Dynamique des applications polynomiales semi-régulières, (French) [Dynamics of semiregular polynomial maps], Ark. Mat., 42 (2004), 61-85.  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), 703-739. 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], Indiana Univ. Math. J., 50 (2001), 881-934. doi: 10.1512/iumj.2001.50.1880.  Google Scholar [7] C. Favre and M. Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup., 40 (2007), 309-349.  Google Scholar [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.  Google Scholar [9] V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$, Amer. J. Math., 124 (2002), 75-106. doi: 10.1353/ajm.2002.0002.  Google Scholar [10] V. Guedj, Dynamics of quadratic polynomial mappings of $\mathbbC^2$, Michigan Math. J., 52 (2004), 627-648. 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), 1275-1296. 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] M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$, Math. Ann., 314 (1999), 403-447. doi: 10.1007/s002080050301.  Google Scholar [14] T. Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan, 46 (1994), 545-555. 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), 153-168. 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), 201-222.  Google Scholar
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