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)

 
 

 

Self-similarity in inverse limit spaces
of the tent family


Authors: Marcy Barge, Karen Brucks and Beverly Diamond
Journal: Proc. Amer. Math. Soc. 124 (1996), 3563-3570
MSC (1991): Primary 54F15, 58F03, 58F12
DOI: https://doi.org/10.1090/S0002-9939-96-03690-8
MathSciNet review: 1363409
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Taking inverse limits of the one-parameter family of tent maps of the interval generates a one-parameter family of inverse limit spaces. We prove that, for a dense set of parameters, these spaces are locally, at most points, the product of a Cantor set and an arc. On the other hand, we show that there is a dense $G_\delta $ set of parameters for which the corresponding space has the property that each neighborhood in the space contains homeomorphic copies of every inverse limit of a tent map.


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

  • 1. M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8(1995), 29-42. MR 95m:58050
  • 2. M. Barge and J. Martin, Endpoints of inverse limit spaces and dynamics, Continua (eds. Cook et al.), Lecture Notes in Pure and Applied Math. 170(1995), Dekker, New York, 165-182. MR 96b:54062
  • 3. M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc, 11(1960), 478-483. MR 22:5959
  • 4. K.M. Brucks and B. Diamond, Monotonicity of auto-expansions, Physica D 51 (1991), 39-42. MR 92h:58101
  • 5. K.M. Brucks, B. Diamond, M.V. Otero-Espinar and C. Tresser, Dense orbits of critical points for the tent map, Contemporary Mathematics, 117 (1991), 57-61. MR 92e:58107
  • 6. P. Collet and J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston, 1980. MR 82j:58078
  • 7. 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 90b:58236
  • 8. S. Holte, Inverse limits of Markov interval maps, preprint.
  • 9. C. Robinson, Dynamical Systems, CRC, Boca Raton, 1995.
  • 10. R.F. Williams, One-dimensional nonwandering sets, Topology 6 (1967), 473-487. MR 36:897

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54F15, 58F03, 58F12

Retrieve articles in all journals with MSC (1991): 54F15, 58F03, 58F12


Additional Information

Marcy Barge
Affiliation: Department of Mathematics, Montana State University, Bozeman, Montana 59717
Email: barge@math.montana.edu

Karen Brucks
Affiliation: Department of Mathematical Sciences, University of Wisconsin at Milwaukee, Milwaukee, Wisconsin 53201
Email: kmbrucks@alpha1.csd.uwm.edu

Beverly Diamond
Affiliation: Department of Mathematics, University of Charleston, Charleston, South Carolina 29424
Email: diamondb@ashley.cofc.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03690-8
Received by editor(s): May 16, 1995
Additional Notes: The first author was supported in part by NSF-DMS-9404145.
Communicated by: Mary Rees
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society