Glasnik Matematicki, Vol. 60, No. 2 (2025), 291-325. \( \)

UNCOUNTABLE FAMILIES OF FANS THAT ADMIT TRANSITIVE HOMEOMORPHISMS

Iztok Banič, Goran Erceg, Judy Kennedy, Chris Mouron and Van Nall

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, SI-2000 Maribor, Slovenia,
Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia,
Andrej Marušič Institute, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia
e-mail:iztok.banic@um.si

Faculty of Science, University of Split, Rudera Boškovića 33, 21000 Split, Croatia
e-mail:goran.erceg@pmfst.hr

Department of Mathematics, Lamar University, 200 Lucas Building, P.O. Box 10047, Beaumont, Texas 77710, USA
e-mail:kennedy9905@gmail.com

Rhodes College, 2000 North Parkway, Memphis, Tennessee 38112, USA
e-mail:mouronc@rhodes.edu

Department of Mathematics, University of Richmond, 221 Richmond Way, Richmond, Virginia 23173, USA
e-mail:vnall@richmond.edu

Abstract.   Recently, we constructed transitive homeomorphisms on the Cantor fan and the Lelek fan. In this paper, we construct a family of uncountably many pairwise non-homeo­morphic smooth fans that admit transitive homeomorphisms. In order to do this, we use our recently developed techniques of combining Mahavier products of closed relations on intervals with quotients of dynamical systems. In addition, we show that the star of Cantor fans admits a transitive homeomorphism. At the end of the paper, we also construct a family of uncountably many pairwise non-homeomorphic non-smooth fans that admit transitive homeomorphisms.

2020 Mathematics Subject Classification.   37B02, 37B45, 54C60, 54F15, 54F17

Key words and phrases.   Closed relations, Mahavier products, transitive dynamical systems, transitive homeomorphisms, smooth fans, Cantor fans, Lelek fans, stars of Cantor fans

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https://doi.org/10.3336/gm.60.2.07

References:

  1. I. Banič, G. Erceg, J. Kennedy, C. Mouron and V. Nall, Transitive mappings on the Cantor fan, Ergodic Theory Dynam. Systems 45 (2025), 2601–2635.
    MathSciNet    CrossRef

  2. I. Banič, G. Erceg and J. Kennedy, A transitive homeomorphism on the Lelek fan, J. Difference Equ. Appl. 29 (2023), 393–418.
    MathSciNet    CrossRef

  3. J. Boroński, P. Minc and S. Štimac, On conjugacy between natural extensions of 1-dimensional maps, to appear in Ergod. Th. Dynam. Sys.
    CrossRef

  4. J. Boroński and P. Oprocha, On dynamics of the Sierpiński carpet, C. R. Math. Acad. Sci. Paris 356 (2018), 340–344.
    MathSciNet    CrossRef

  5. W.  D. Bula and L. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), 529–534.
    MathSciNet    CrossRef

  6. W. J. Charatonik, The Lelek fan is unique, Houston J. Math. 15 (1989), 27–34.
    MathSciNet

  7. J. J. Charatonik, On fans, Dissertationes Math. (Rozprawy Mat.) 7133 (1967), 37 pp.

  8. J. Činč and P. Oprocha, Parametrized family of pseudo-arc attractors: physical measures and prime end rotations, Proc. London Math. Soc. (3) 125 (2022), 318–357.
    MathSciNet    CrossRef

  9. R. Engelking, General topology, Heldermann, Berlin, 1989.
    MathSciNet

  10. L. C. Hoehn and C. Mouron, Hierarchies of chaotic maps on continua, Ergodic Theory Dynam. Systems 34 (2014), 1897–1913.
    MathSciNet    CrossRef

  11. J. Kennedy, A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map, Trans. Amer. Math. Soc. 326 (1991), 773–793.
    MathSciNet    CrossRef

  12. A. Lelek, On plane dendroids and their end-points in the classical sense, Fund. Math. 49 (1960/1961), 301–319.
    MathSciNet    CrossRef

  13. T. Maćkowiak, Semi-confluent mappings and their invariants, Fund. Math. 79 (1973), 251–264.
    MathSciNet    CrossRef

  14. P. Minc and W. R. R. Transue, A transitive map on \([0,1]\) whose inverse limit is the pseudoarc, Proc. Amer. Math. Soc. 111 (1991), 1165–1170.
    MathSciNet    CrossRef

  15. C. Mouron, Expansive homeomorphisms and indecomposable subcontinua, Topology Appl. 126 (2002), 13–28.
    MathSciNet    CrossRef

  16. C. Mouron, Tree-like continua do not admit expansive homeomorphisms, Proc. Amer. Math. Soc. 130 (2002), 3409–3413.
    MathSciNet    CrossRef

  17. J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, 1975.
    MathSciNet

  18. S. B. Nadler, Continuum theory, Marcel Dekker, Inc., New York, 1992.
    MathSciNet

  19. P. Oprocha, Lelek fan admits completely scrambled weakly mixing homeomorphism, Bull. Lond. Math. Soc. 57 (2025), 432–443.
    MathSciNet

  20. S. Willard, General topology, Dover Publications, New York, 2004.
    MathSciNet
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