Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Shadowing, entropy and a homeomorphism of the pseudoarc

Authors: Piotr Koscielniak and Piotr Oprocha
Journal: Proc. Amer. Math. Soc. 138 (2010), 1047-1057
MSC (2000): Primary 37B45; Secondary 54H20, 37B40, 37B05
Published electronically: November 10, 2009
MathSciNet review: 2566570
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we provide a method of constructing continuous maps $ f\colon [0,1]\rightarrow [0,1]$ such that $ f$ is topologically mixing, has the shadowing property, and the inverse limit of copies of $ [0,1]$ with $ f$ as the bonding map is the pseudoarc. Such a map can be obtained as an arbitrarily small $ \mathcal{C}^0$-perturbation of any topologically exact map on $ [0,1]$. We have therefore answered, in the affirmative, a question posed by Chen and Li in 1993.

References [Enhancements On Off] (What's this?)

  • 1. L. Alsedà, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, second ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. MR 1807264 (2001j:37073)
  • 2. M. Barge, A method for constructing attractors, Ergodic Theory Dynam. Systems 8 (1988), 331-349. MR 961734 (90a:58100)
  • 3. M. Barge and J. Kennedy, Continuum theory and topological dynamics, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 633-644. MR 1078669
  • 4. M. Barge and J. Martin, Dense periodicity on the interval, Proc. Amer. Math. Soc. 94 (1985), 731-735. MR 792293 (87b:58068)
  • 5. R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43-51. MR 0043451 (13:265b)
  • 6. L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300 (1987), 297-306. MR 871677 (88c:58032)
  • 7. L. Block, J. Keesling, and V. V. Uspenskij, Inverse limits which are the pseudoarc, Houston J. Math. 26 (2000), 629-638. MR 1823960 (2002b:54040)
  • 8. L. S. Block and W. A. Coppel, Dynamics in one dimension, Lecture Notes in Mathematics, vol. 1513, Springer-Verlag, Berlin, 1992. MR 1176513 (93g:58091)
  • 9. A. M. Blokh, The ``spectral'' decomposition for one-dimensional maps, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 1-59. MR 1346496 (96e:58087)
  • 10. R. Bowen, Topological entropy and axiom A, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, RI, 1970, pp. 23-41. MR 0262459 (41:7066)
  • 11. L. Chen and S. H. Li, Shadowing property for inverse limit spaces, Proc. Amer. Math. Soc. 115 (1992), 573-580. MR 1097338 (92i:58094)
  • 12. L. Chen and S. H. Li, Dynamical connections between a continuous map and its inverse limit space, Continuum theory and dynamical systems, Lecture Notes in Pure and Appl. Math., vol. 149, Dekker, New York, 1993, pp. 89-97. MR 1235348 (94d:54089)
  • 13. E. M. Coven, I. Kan, and J. A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), 227-241. MR 946440 (90b:58236)
  • 14. M. Denker, C. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin, 1976. MR 0457675 (56:15879)
  • 15. G. W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J. 31 (1964), 421-425. MR 0166766 (29:4039)
  • 16. J. Kennedy, A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map, Trans. Amer. Math. Soc. 326 (1991), 773-793. MR 1010412 (91k:54062)
  • 17. B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
  • 18. D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33 (2007), no. 1, 76-86. MR 2301847 (2008b:37024)
  • 19. S. Macıas, Topics on continua, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2147759 (2006f:54035)
  • 20. 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. MR 1042271 (91g:54050)
  • 21. E. E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc. 63 (1948), 581-594. MR 0025733 (10:56i)
  • 22. C. Mouron, Entropy of shift maps of the pseudo-arc, preprint.
  • 23. O. M. Šarkovs$ '$kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Z. 16 (1964), 61-71. MR 0159905 (28:3121)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37B45, 54H20, 37B40, 37B05

Retrieve articles in all journals with MSC (2000): 37B45, 54H20, 37B40, 37B05

Additional Information

Piotr Koscielniak
Affiliation: Institute of Mathematics of the Jagiellonian University, ul. Lojasiewicza 6, 30-348 Kraków, Poland

Piotr Oprocha
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain – and – Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

Keywords: Pseudoarc, shadowing, entropy, crooked map, topological mixing
Received by editor(s): May 4, 2009
Received by editor(s) in revised form: August 4, 2009
Published electronically: November 10, 2009
Communicated by: Bryna Kra
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society