American Institute of Mathematical Sciences

Advanced
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

[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 polynomial skew products on $\mathbb{C}^2$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2283-2305. doi: 10.3934/dcds.2014.34.2283
[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

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences