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Conformal Geometry and Dynamics

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Dynamical properties of families of holomorphic mappings


Authors: Ratna Pal and Kaushal Verma
Journal: Conform. Geom. Dyn. 19 (2015), 323-350
MSC (2010): Primary 37F45
DOI: https://doi.org/10.1090/ecgd/285
Published electronically: December 10, 2015
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Abstract: We study some dynamical properties of skew products of Hénon maps of $ \mathbb{C}^2$ that are fibered over a compact metric space $ M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. This analogy is carried forth in two ways: it is shown that the successive pull-backs of a suitable current by the skew Hénon maps converges to a multiple of the fibered stable current and secondly, this convergence result is used to obtain a lower bound on the topological entropy of the skew product in some special cases. The other class of maps that are studied are skew products of holomorphic endomorphisms of $ \mathbb{P}^k$ that are again fibered over a compact base. We define the fibered Fatou component and show that they are pseudoconvex and Kobayashi hyperbolic.


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  • [1] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ {\bf C}^2$: currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), no. 1, 69-99. MR 1079840 (92a:32035), https://doi.org/10.1007/BF01239509
  • [2] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ {\bf C}^2$. II. Stable manifolds and recurrence, J. Amer. Math. Soc. 4 (1991), no. 4, 657-679. MR 1115786 (92m:32048), https://doi.org/10.2307/2939284
  • [3] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ \mathbf {C}^2$. III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann. 294 (1992), no. 3, 395-420. MR 1188127 (93k:32062), https://doi.org/10.1007/BF01934331
  • [4] Dan Coman and John Erik Fornæss, Green's functions for irregular quadratic polynomial automorphisms of $ {\bf C}^3$, Michigan Math. J. 46 (1999), no. 3, 419-459. MR 1721516 (2000i:32030), https://doi.org/10.1307/mmj/1030132474
  • [5] Dan Coman and Vincent Guedj, Invariant currents and dynamical Lelong numbers, J. Geom. Anal. 14 (2004), no. 2, 199-213. MR 2051683 (2005d:32056), https://doi.org/10.1007/BF02922068
  • [6] Henry de Thélin, Endomorphismes pseudo-aléatoires dans les espaces projectifs I, Manuscripta Math. 142 (2013), no. 3-4, 347-367 (French, with English and French summaries). MR 3117166, https://doi.org/10.1007/s00229-012-0603-9
  • [7] Henry De Thélin, Endomorphismes pseudo-aléatoires dans les espaces projectifs II, J. Geom. Anal. 25 (2015), no. 1, 204-225 (French, with English and French summaries). MR 3299276, https://doi.org/10.1007/s12220-013-9422-9
  • [8] T. C. Dinh, N. Sibony, Rigidity of Julia sets for Hénon type maps, Proceedings of the 2008-2011 Summer Institute at Bedlewo. Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory, To appear.
  • [9] Charles Favre and Mattias Jonsson, Dynamical compactifications of $ {\bf C}^2$, Ann. of Math. (2) 173 (2011), no. 1, 211-248. MR 2753603 (2012d:32025), https://doi.org/10.4007/annals.2011.173.1.6
  • [10] Shmuel Friedland and John Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 67-99. MR 991490 (90f:58163), https://doi.org/10.1017/S014338570000482X
  • [11] John Erik Fornæss and Nessim Sibony, Complex Hénon mappings in $ {\bf C}^2$ and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), no. 2, 345-380. MR 1150591 (93d:32040), https://doi.org/10.1215/S0012-7094-92-06515-X
  • [12] John Erik Fornæss and He Wu, Classification of degree $ 2$ polynomial automorphisms of $ {\bf C}^3$, Publ. Mat. 42 (1998), no. 1, 195-210. MR 1628170 (99e:14015), https://doi.org/10.5565/PUBLMAT_42198_10
  • [13] Vincent Guedj, Courants extrémaux et dynamique complexe, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 3, 407-426 (French, with English and French summaries). MR 2166340 (2006k:32070), https://doi.org/10.1016/j.ansens.2005.02.001
  • [14] Vincent Guedj and Nessim Sibony, Dynamics of polynomial automorphisms of $ \mathbf {C}^k$, Ark. Mat. 40 (2002), no. 2, 207-243. MR 1948064 (2004b:32029), https://doi.org/10.1007/BF02384535
  • [15] John H. Hubbard and Peter Papadopol, Superattractive fixed points in $ {\bf C}^n$, Indiana Univ. Math. J. 43 (1994), no. 1, 321-365. MR 1275463 (95e:32025), https://doi.org/10.1512/iumj.1994.43.43014
  • [16] Mattias Jonsson, Dynamics of polynomial skew products on $ \mathbf {C}^2$, Math. Ann. 314 (1999), no. 3, 403-447. MR 1704543 (2000f:32025), https://doi.org/10.1007/s002080050301
  • [17] Mattias Jonsson, Ergodic properties of fibered rational maps, Ark. Mat. 38 (2000), no. 2, 281-317. MR 1785403 (2002k:37073), https://doi.org/10.1007/BF02384321
  • [18] Han Peters, Non-autonomous dynamics in $ \mathbb{P}^k$, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1295-1304. MR 2158406 (2006b:37085), https://doi.org/10.1017/S0143385704000987
  • [19] Han Peters and Erlend Fornæss Wold, Non-autonomous basins of attraction and their boundaries, J. Geom. Anal. 15 (2005), no. 1, 123-136. MR 2132268 (2006a:37035), https://doi.org/10.1007/BF02921861
  • [20] Nessim Sibony, Dynamique des applications rationnelles de $ \mathbf {P}^k$, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix-x, xi-xii, 97-185 (French, with English and French summaries). MR 1760844 (2001e:32026)
  • [21] John Smillie, The entropy of polynomial diffeomorphisms of $ {\bf C}^2$, Ergodic Theory Dynam. Systems 10 (1990), no. 4, 823-827. MR 1091429 (92b:58131), https://doi.org/10.1017/S0143385700005927
  • [22] Tetsuo Ueda, Fatou sets in complex dynamics on projective spaces, J. Math. Soc. Japan 46 (1994), no. 3, 545-555. MR 1276837 (95d:32030), https://doi.org/10.2969/jmsj/04630545
  • [23] S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000. Translated from the 1995 Japanese original and revised by the authors. MR 1747010 (2002c:37064)
  • [24] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285-300. MR 889979 (90g:58008), https://doi.org/10.1007/BF02766215
  • [25] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108 (84e:28017)

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Additional Information

Ratna Pal
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: ratna10@math.iisc.ernet.in

Kaushal Verma
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
Email: kverma@math.iisc.ernet.in

DOI: https://doi.org/10.1090/ecgd/285
Received by editor(s): April 22, 2015
Received by editor(s) in revised form: September 28, 2015
Published electronically: December 10, 2015
Additional Notes: The first named author was supported by CSIR-UGC (India) fellowship
The second named author was supported by the DST Swarna Jayanti Fellowship 2009–2010 and a UGC–CAS Grant
Article copyright: © Copyright 2015 American Mathematical Society

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