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On indecomposability in chaotic attractors


Authors: Jan P. Boroński and Piotr Oprocha
Journal: Proc. Amer. Math. Soc. 143 (2015), 3659-3670
MSC (2010): Primary 54F15
DOI: https://doi.org/10.1090/S0002-9939-2015-12526-9
Published electronically: March 25, 2015
MathSciNet review: 3348807
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Abstract: We exhibit a Li-Yorke chaotic interval map $ F$ such that the inverse limit $ X_F=\varprojlim \{F,[0,1]\}$ does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if $ \varphi $ has positive entropy then $ X_\varphi $ contains an indecomposable subcontinuum. Each subcontinuum of $ X_F$ is homeomorphic to one of the following: an arc, or $ X_F$, or a topological ray limiting to $ X_F$. Through our research, we found that it follows that $ X_F$ is a chaotic attractor of a planar homeomorphism. In addition, $ F$ can be modified to give a cofrontier that is a chaotic attractor of a planar homeomorphism but contains no indecomposable subcontinuum. Finally, $ F$ can be modified, without removing or introducing new periods, to give a chaotic zero entropy interval map, such that the corresponding inverse limit contains the pseudoarc.


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  • [1] Ethan Akin, Joseph Auslander, and Kenneth Berg, When is a transitive map chaotic?, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 25-40. MR 1412595 (97i:58106)
  • [2] Lluís Alsedà, Jaume Llibre, and Michał Misiurewicz, 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 (2001j:37073)
  • [3] Marcy Barge and Joe Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289 (1985), no. 1, 355-365. MR 779069 (86h:58079), https://doi.org/10.2307/1999705
  • [4] Marcy Barge and Joe Martin, The construction of global attractors, Proc. Amer. Math. Soc. 110 (1990), no. 2, 523-525. MR 1023342 (90m:58123), https://doi.org/10.2307/2048099
  • [5] Marcy Barge and Robert Roe, Circle maps and inverse limits, Topology Appl. 36 (1990), no. 1, 19-26. MR 1062181 (91f:58071), https://doi.org/10.1016/0166-8641(90)90032-W
  • [6] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742. MR 0027144 (10,261a)
  • [7] R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. MR 0043451 (13,265b)
  • [8] L. Block, J. Keesling, and V. V. Uspenskij, Inverse limits which are the pseudoarc, Houston J. Math. 26 (2000), no. 4, 629-638. MR 1823960 (2002b:54040)
  • [9] J. S. Canovas, On topological sequence entropy and chaotic maps on inverse limit spaces, Acta Math. Univ. Comenian. (N.S.) 68 (1999), no. 2, 205-211. MR 1757789 (2001a:37042)
  • [10] Jean-Paul Delahaye, Fonctions admettant des cycles d'ordre n'importe quelle puissance de $ 2$ et aucun autre cycle, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 4, A323-A325 (French, with English summary). MR 591762 (83e:58073a)
  • [11] Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376 (91a:58114)
  • [12] Bau-Sen Du, A simple proof of Sharkovsky's theorem revisited, Amer. Math. Monthly 114 (2007), no. 2, 152-155. MR 2290366 (2007j:37063)
  • [13] George W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J. 31 (1964), 421-425. MR 0166766 (29 #4039)
  • [14] W. T. Ingram, Periodic points for homeomorphisms of hereditarily decomposable chainable continua, Proc. Amer. Math. Soc. 107 (1989), no. 2, 549-553. MR 984796 (90f:54052), https://doi.org/10.2307/2047846
  • [15] W. T. Ingram, Periodicity and indecomposability, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1907-1916. MR 1283553 (95j:58087), https://doi.org/10.2307/2161010
  • [16] Carlos Islas, A 2-equivalent Kelley continuum, Glas. Mat. Ser. III 46(66) (2011), no. 1, 249-268. MR 2810939 (2012j:54064), https://doi.org/10.3336/gm.46.1.18
  • [17] Judy Kennedy, A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map, Trans. Amer. Math. Soc. 326 (1991), no. 2, 773-793. MR 1010412 (91k:54062), https://doi.org/10.2307/2001783
  • [18] Wayne Lewis, Most maps of the pseudo-arc are homeomorphisms, Proc. Amer. Math. Soc. 91 (1984), no. 1, 147-154. MR 735582 (85g:54025), https://doi.org/10.2307/2045287
  • [19] Tien Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985-992. MR 0385028 (52 #5898)
  • [20] Piotr Minc and W. R. R. Transue, Accessible points of hereditarily decomposable chainable continua, Trans. Amer. Math. Soc. 332 (1992), no. 2, 711-727. MR 1073777 (92j:54045), https://doi.org/10.2307/2154192
  • [21] Piotr Minc and W. R. R. Transue, A transitive map on $ [0,1]$ whose inverse limit is the pseudoarc, Proc. Amer. Math. Soc. 111 (1991), no. 4, 1165-1170. MR 1042271 (91g:54050), https://doi.org/10.2307/2048584
  • [22] Jerzy Mioduszewski, Wykłady z topologii, Prace Naukowe Uniwersytetu Ślaskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 1885, Wydawnictwo Uniwersytetu Ślaskiego, Katowice, 2003 (Polish, with English and Russian summaries). Zbiory spójne i kontinua. [Connected sets and continua]. MR 2046190 (2005a:54001)
  • [23] M. Misiurewicz, Horseshoes for continuous mappings of an interval, Dynamical systems (Bressanone, 1978) Liguori, Naples, 1980, pp. 125-135. MR 660643 (83h:58076)
  • [24] M. Misiurewicz and J. Smítal, Smooth chaotic maps with zero topological entropy, Ergodic Theory Dynam. Systems 8 (1988), no. 3, 421-424. MR 961740 (90a:58118), https://doi.org/10.1017/S0143385700004557
  • [25] Christopher Mouron, Entropy of shift maps of the pseudo-arc, Topology Appl. 159 (2012), no. 1, 34-39. MR 2852946 (2012k:37040), https://doi.org/10.1016/j.topol.2011.07.014
  • [26] Christopher Mouron, Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2783-2791. MR 2801619 (2012e:37035), https://doi.org/10.1090/S0002-9939-2010-10783-9
  • [27] 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 (93m:54002)
  • [28] RUETTE, S. Chaos for continuous interval maps: a survey of relationship between the various sorts of chaos, Unpublished manuscript available at "http://www.math.u-psud.fr/ ruette/".
  • [29] J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), no. 1, 269-282. MR 849479 (87m:58107), https://doi.org/10.2307/2000468
  • [30] Xiang Dong Ye, The dynamics of homeomorphisms of hereditarily decomposable chainable continua, Topology Appl. 64 (1995), no. 1, 85-93. MR 1339760 (96g:54048), https://doi.org/10.1016/0166-8641(94)00091-G

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Additional Information

Jan P. Boroński
Affiliation: National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic — and — AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: jan.boronski@osu.cz

Piotr Oprocha
Affiliation: AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland — and — Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
Email: oprocha@agh.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2015-12526-9
Keywords: Interval map, arc-like, Li-Yorke chaotic, indecomposable continuum
Received by editor(s): March 29, 2013
Received by editor(s) in revised form: February 14, 2014
Published electronically: March 25, 2015
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

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