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

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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 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.

**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**

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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