Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T20:53:34.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Universal Entire Functions That Define Order Isomorphisms of Countable Real Sets

Part of: Universal holomorphic functions Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  10 April 2019

P. M. Gauthier
P. M. Gauthier*
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP-6128 Centreville, Montréal, H3C3J7 Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism that is the restriction of a universal entire function.

Keywords

universal function

MSC classification

Primary: 30E10: Approximation in the complex domain
Secondary: 30K20: Compositional universality
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 767 - 779
Copyright
© Canadian Mathematical Society 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported by NSERC (Canada) grant RGPIN-2016-04107.

References

Bagchi, B., Recurrence in topological dynamics and the Riemann hypothesis . Acta Math. Hungar. 50(1987), no. 3–4, 227–240. https://doi.org/10.1007/BF01903937 Google Scholar
Barth, K. F. and Schneider, W. J., Entire functions mapping countable dense subsets of the reals onto each other monotonically . J. London Math. Soc. (2) 2(1970), 620–626. https://doi.org/10.1112/jlms/2.Part_4.620 Google Scholar
Barth, K. F. and Schneider, W. J., Entire functions mapping arbitrary countable dense sets and their complements onto each other . J. London Math. Soc. (2) 4(1971/72), 482–488. https://doi.org/10.1112/jlms/s2-4.3.482 Google Scholar
Bayart, F. and Matheron, E., Dynamics of linear operators . Cambridge Univ. Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511581113 Google Scholar
Birkhoff, G. D., Démonstration d’un théorème élémentaire sur les fonctions entières . C.R. Acad. Sci. Paris 189(1929), 473–475.Google Scholar
Burke, M. R., Simultaneous approximation and interpolation of increasing functions by increasing entire functions. (English summary) J. Math. Anal. Appl. 350(2009), no. 2, 845–858. https://doi.org/10.1016/j.jmaa.2008.08.018 Google Scholar
Burke, M. R., Entire functions mapping uncountable dense sets of reals onto each other monotonically . Trans. Amer. Math. Soc. 361(2009), no. 6, 2871–2911. https://doi.org/10.1090/S0002-9947-09-04924-1 Google Scholar
Burke, M. R., Generic approximation and interpolation by entire functions via restriction of the values of the derivatives. (2017). Summer Conference on Topology and Its Applications. 44. http://ecommons.udayton.edu/topology_conf/44 Google Scholar
Deutsch, F., Simultaneous interpolation and approximation in topological linear spaces . SIAM J. Appl. Math. 14(1966), 1180–1190. https://doi.org/10.1137/0114095 Google Scholar
Erdős, P., Some unsolved problems . Michigan Math. J. 4(1957), 291–300.Google Scholar
Franklin, P., Analytic transformations of everywhere dense point sets . Trans. Amer. Math. Soc. 27(1925), no. 1, 91–100. https://doi.org/10.2307/1989166 Google Scholar
Grosse-Erdmann, K.-G. and Peris Manguillot, A., Linear chaos . Springer, London, 2011. https://doi.org/10.1007/978-1-4471-2170-1 Google Scholar
Maurer, W. D., Conformal equivalence of countable dense sets . Proc. Amer. Math. Soc. 18(1967), 269–270. https://doi.org/10.2307/2035276 Google Scholar
Morayne, M., Measure preserving analytic diffeomorphisms of countable dense sets in ℂ n and ℝ n . Colloq. Math. 52(1987), no. 1, 93–98. https://doi.org/10.4064/cm-52-1-93-98 Google Scholar
Pietroń, M., Measure-preserving countable dense homogeneity of the Hilbert cube . Topology Appl. 160(2013), no. 2, 257–263. https://doi.org/10.1016/j.topol.2012.10.006 Google Scholar
Rosay, J.-P. and Rudin, W., Holomorphic maps from C n to C n . Trans. Amer. Math. Soc. 310(1988), no. 1, 47–86. https://doi.org/10.2307/2001110 Google Scholar
Stäckel, P., Ueber arithmetische Eigenschaften analytischer Funktionen . Math. Ann. 46(1895), 513–520.Google Scholar
Voronin, S. M., A theorem on the “universality” of the Riemann zeta-function. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 39(1975), no. 3, 475–486. 703.Google Scholar