On almost one-to-one maps
Authors:
Alexander Blokh, Lex Oversteegen and E. D. Tymchatyn
Journal:
Trans. Amer. Math. Soc. 358 (2006), 5003-5014
MSC (2000):
Primary 57N35, 54C10; Secondary 37B45
DOI:
https://doi.org/10.1090/S0002-9947-06-03922-5
Published electronically:
June 13, 2006
MathSciNet review:
2231882
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A continuous map of topological spaces
is said to be almost
-to-
if the set of the points
such that
is dense in
; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and
-compact spaces (e.g.,
-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if
is a minimal self-mapping of a 2-manifold
, then point preimages under
are tree-like continua and either
is a union of 2-tori, or
is a union of Klein bottles permuted by
.
- 1. L. Alseda, M. Misiurewicz, and J. Llibre, Combinational Dynamics and Entropy in Dimension One, 2nd edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific, Singapore (2001). MR 1807264 (2001j:37073)
- 2. J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. Journ. 32 (1980), pp. 177-188.MR 0580273 (82b:58049)
- 3. L. Block and W. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics 1513, Springer-Verlag, New York (1992).MR 1176513 (93g:58091)
- 4. A. M. Blokh, L. G. Oversteegen, and E.D. Tymchatyn, On minimal maps of 2-manifolds, Erg. Th. and Dyn. Syst. 25 (2005), 41-57. MR 2122911
- 5. A. M. Blokh, L. G. Oversteegen, and E.D. Tymchatyn, Applications of almost one-to-one maps, Topology and Appl. 153 (2006), 1571-1585.
- 6. R. Engelking, Dimension theory, North-Holland and PWN (1978).MR 0482697 (58:2753b)
- 7. S. Kolyada, L. Snoha and S. Trofimchuk, Noninvertible minimal maps, Fundamenta Mathematicae 168 (2001), 141-163.MR 1852739 (2002j:37017)
- 8. J. van Mill, Infinite Dimensional Topology; Prerequisite and Introduction, North-Holland, Amsterdam (1989). MR 0977744 (90a:57025)
- 9. S. B. Nadler, Jr., Continuum theory, Marcel Dekker Inc., New York (1992).MR 1192552 (93m:54002)
- 10. M. Rees, A point distal transformation of the torus, Israel J. Math. 32 (1979), 201-208. MR 0531263 (81g:54054)
- 11. G. T. Whyburn, Analytic topology, vol. 28, AMS Coll. Publications, Providence, RI, 1942. MR 0007095 (4:86b)
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Additional Information
Alexander Blokh
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email:
ablokh@math.uab.edu
Lex Oversteegen
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email:
overstee@math.uab.edu
E. D. Tymchatyn
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
Email:
tymchat@snoopy.math.usask.ca
DOI:
https://doi.org/10.1090/S0002-9947-06-03922-5
Keywords:
Almost one-to-one map,
embedding,
homeomorphism,
light map
Received by editor(s):
February 29, 2004
Received by editor(s) in revised form:
October 21, 2004
Published electronically:
June 13, 2006
Additional Notes:
The first author was partially supported by NSF Grant DMS-0140349
The second author was partially supported by NSF Grant DMS-0072626
The third author was partially supported by NSERC grant OGP005616
Article copyright:
© Copyright 2006
American Mathematical Society