# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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], 50 (2001), 881. doi: 10.1512/iumj.2001.50.1880. Google Scholar [6] C. Favre and M. Jonsson, The Valuative Tree,, Lecture Notes in Mathematics, (1853). doi: 10.1007/b100262. Google Scholar [7] C. Favre and M. Jonsson, Eigenvaluations,, Ann. Sci. École Norm. Sup. (4), 40 (2007), 309. 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. 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, (1992), 135. Google Scholar [10] V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$,, Amer. J. Math., 124 (2002), 75. 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. 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] 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. doi: 10.4064/fm210-1-4. Google Scholar [14] M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$,, Math. Ann., 314 (1999), 403. 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. 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. 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. 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. 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. 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. 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. 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. 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], 50 (2001), 881. doi: 10.1512/iumj.2001.50.1880. Google Scholar [6] C. Favre and M. Jonsson, The Valuative Tree,, Lecture Notes in Mathematics, (1853). doi: 10.1007/b100262. Google Scholar [7] C. Favre and M. Jonsson, Eigenvaluations,, Ann. Sci. École Norm. Sup. (4), 40 (2007), 309. 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. 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, (1992), 135. Google Scholar [10] V. Guedj, Dynamics of polynomial mappings of $\mathbbC^2$,, Amer. J. Math., 124 (2002), 75. 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. 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] 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. doi: 10.4064/fm210-1-4. Google Scholar [14] M. Jonsson, Dynamics of polynomial skew products on $\mathbbC^2$,, Math. Ann., 314 (1999), 403. 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. 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. 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. 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. doi: 10.2996/kmj/1341401055. Google Scholar [19] K. Ueno, Polynomial skew products whose Julia sets have infinitely many symmetries,, submitted., (). Google Scholar
 [1] Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219 [2] Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487 [3] Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098 [4] Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007 [5] Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791 [6] Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307 [7] 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 [8] Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 1-42. doi: 10.3934/jmd.2018012 [9] P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883 [10] Saša Kocić. Reducibility of skew-product systems with multidimensional Brjuno base flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 261-283. doi: 10.3934/dcds.2011.29.261 [11] Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081 [12] Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873 [13] Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767 [14] Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657 [15] Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó, Joan Carles Tatjer. Classification of linear skew-products of the complex plane and an affine route to fractalization. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3767-3787. doi: 10.3934/dcds.2019153 [16] Ali Unver, Christian Ringhofer, Dieter Armbruster. A hyperbolic relaxation model for product flow in complex production networks. Conference Publications, 2009, 2009 (Special) : 790-799. doi: 10.3934/proc.2009.2009.790 [17] Juan A. Calzada, Rafael Obaya, Ana M. Sanz. Continuous separation for monotone skew-product semiflows: From theoretical to numerical results. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 915-944. doi: 10.3934/dcdsb.2015.20.915 [18] Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291 [19] Bogdan Sasu, A. L. Sasu. Input-output conditions for the asymptotic behavior of linear skew-product flows and applications. Communications on Pure & Applied Analysis, 2006, 5 (3) : 551-569. doi: 10.3934/cpaa.2006.5.551 [20] Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

2018 Impact Factor: 1.143