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Iteration of Analytic Multifunctions

Published online by Cambridge University Press:  22 January 2016

Maciej Klimek
Maciej Klimek*
Affiliation:
Department of Mathematics, Uppsala University, P.O.Box 480, 751-06 Uppsala, Sweden, [email protected]
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Abstract

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It is shown that iteration of analytic set-valued functions can be used to generate composite Julia sets in CN. Then it is shown that the composite Julia sets generated by a finite family of regular polynomial mappings of degree at least 2 in CN, depend analytically on the generating polynomials, in the sense of the theory of analytic set-valued functions. It is also proved that every pluriregular set can be approximated by composite Julia sets. Finally, iteration of infinitely many polynomial mappings is used to give examples of pluriregular sets which are not composite Julia sets and on which Markov’s inequality fails.

Keywords

32F0532H5030D0558F23
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 162 , June 2001 , pp. 19 - 40
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Baribeau, L. and Ransford, T. J., Meromorphic multifunctions in complex dynamics, Ergodic Theory Dynamical Systems, 12 (1992), 39–52.Google Scholar
[2] Douady, A., Does Julia set depend continuously on the polynomial, Proc. Symp. Appl. Math., 49 (1994), 91–138.Google Scholar
[3] Hutchinson, J. E., Fractals and self similarity, Indiana Univ. Math. J., 30 (1981), 713–747.Google Scholar
[4] Klimek, M., A note on L-regularity of compact sets in Cn, Bull. Acad. Pol. Sci., Série Sci. Mat., 29 (1981), 449–451.Google Scholar
[5] Klimek, M., Extremal plurisubharmonic functions and L-regular sets in Cn, Proc. Roy. Irish Acad. Sect. A, 82A (1982), 217–230.Google Scholar
[6] Klimek, M., Joint Spectra and Analytic Set-valued Functions, Trans. Amer. Math. Soc, 294 (1986), 187–196.Google Scholar
[7] Klimek, M., A Criterion of Analyticity for Set-valued Functions, Proc. Roy. Irish Acad. Sect. A, 86A (1986), 1–4.Google Scholar
[8] Klimek, M., Pluripotential Theory, Oxford University Press, Oxford- New York - Tokyo, 1991.Google Scholar
[9] Klimek, M., Metrics associated with extremal plurisubharmonic functions, Proc. Amer. Math. Soc, 123 (1995), 2763–2770.Google Scholar
[10] Klimek, M., Inverse iteration systems in CN , In: Complex analysis and differential equations (Proceedings of the Marcus Wallenberg Symposium in Honour of Matts Essén, Held in Uppsala, Sweden, June 15–18, 1997), 206–214, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 64, Uppsala Univ., Uppsala, 1999.Google Scholar
[11] Kosek, M., Holder Continuity Property of filled-in Julia sets in Cn, Proc. Amer. Math. Soc, 125 (1997), 2029–2032.Google Scholar
[12] Klimek, M., Holder Continuity Property of composite Julia sets, Bull. Polish Acad. Sci. Math., 46(4) (1998), 391–399.Google Scholar
[13] Thanh Van, Nguyen and Plesniak, W., Invariance of L-regularity and Leja’s polynomial condition under holomorphic mappings, Proc. Roy. Irish Acad. Sect. A, 84A (1984), 111–115.Google Scholar
[14] Plesniak, W., Invariance of the L-regularity of compact sets in Cn, Trans. Amer. Math Soc, 246 (1978), 373–383.Google Scholar
[15] Plesniak, W., A Cantor regular set which does not have Markov’s property, Ann. Polon. Math. vol., LI (1990), 269–274.Google Scholar
[16] Plesniak, W., Markov’s inequality and the existence of an extension operator for C∞ functions, J. Approx. Theory,, 61 (1990), 106–117.Google Scholar
[17] Ransford, T. J., Analytic Multivalued Functions, Essay presented to the Trinity College Research Fellowship Competition, Cambridge University, 1983.Google Scholar
[18] Ransford, T. J., Open mapping, inversion and implicit function theorems for analytic multi valued functions, Proc. London Math. Soc, 49 (1984), 537–562.Google Scholar
[19] Sadullaev, A., Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys, 36:4 (1981), 61–119.Google Scholar
[20] Siciak, J., On metrics associated with extremal plurisubharmonic functions, Bull. Pol. Acad. Sci. (Mathematics), 45 (1997), 151–161.Google Scholar
[21] Slodkowski, Z., Analytic set-valued functions and spectra, Math. Ann., 256 (1981), 363–386.Google Scholar
[22] Slodkowski, Z., An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, Trans. Amer. Math. Soc, 294 (1986), 367–377.Google Scholar
[23] Slodkowski, Z., Analytic perturbation of the Taylor spectrum, Trans. Amer. Math. Soc, 297 (1986), 319–336.Google Scholar