## On almost one-to-one maps

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- by Alexander Blokh, Lex Oversteegen and E. D. Tymchatyn PDF
- Trans. Amer. Math. Soc.
**358**(2006), 5003-5014 Request permission

## Abstract:

A continuous map $f:X\to Y$ of topological spaces $X, Y$ is said to be*almost $1$-to-$1$*if the set of the points $x\in X$ such that $f^{-1}(f(x))=\{x\}$ is dense in $X$; it is said to be

*light*if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and $\sigma$-compact spaces (e.g., $n$-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 $f$ is a minimal self-mapping of a 2-manifold $M$, then point preimages under $f$ are tree-like continua and either $M$ is a union of 2-tori, or $M$ is a union of Klein bottles permuted by $f$.

## References

- Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz,
*Combinatorial dynamics and entropy in dimension one*, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR**1807264**, DOI 10.1142/4205 - Joseph Auslander and James A. Yorke,
*Interval maps, factors of maps, and chaos*, Tohoku Math. J. (2)**32**(1980), no. 2, 177–188. MR**580273**, DOI 10.2748/tmj/1178229634 - L. S. Block and W. A. Coppel,
*Dynamics in one dimension*, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR**1176513**, DOI 10.1007/BFb0084762 - Alexander Blokh, Lex Oversteegen, and E. D. Tymchatyn,
*On minimal maps of 2-manifolds*, Ergodic Theory Dynam. Systems**25**(2005), no. 1, 41–57. MR**2122911**, DOI 10.1017/S0143385704000331 - A. M. Blokh, L. G. Oversteegen, and E.D. Tymchatyn,
*Applications of almost one-to-one maps*, Topology and Appl.**153**(2006), 1571–1585. - Ryszard Engelking,
*Teoria wymiaru*, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR**0482696** - Sergiĭ Kolyada, L’ubomír Snoha, and Sergeĭ Trofimchuk,
*Noninvertible minimal maps*, Fund. Math.**168**(2001), no. 2, 141–163. MR**1852739**, DOI 10.4064/fm168-2-5 - J. van Mill,
*Infinite-dimensional topology*, North-Holland Mathematical Library, vol. 43, North-Holland Publishing Co., Amsterdam, 1989. Prerequisites and introduction. MR**977744** - Sam B. Nadler Jr.,
*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552** - M. Rees,
*A point distal transformation of the torus*, Israel J. Math.**32**(1979), no. 2-3, 201–208. MR**531263**, DOI 10.1007/BF02764916 - Gordon Thomas Whyburn,
*Analytic Topology*, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR**0007095**, DOI 10.1090/coll/028

## Additional Information

**Alexander Blokh**- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 196866
- Email: ablokh@math.uab.edu
**Lex Oversteegen**- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 134850
- Email: overstee@math.uab.edu
**E. D. Tymchatyn**- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6
- MR Author ID: 175580
- Email: tymchat@snoopy.math.usask.ca
- 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 - © Copyright 2006 American Mathematical Society
- 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
- MathSciNet review: 2231882