Self-similarity in inverse limit spaces of the tent family

Barge, Marcy; Brucks, Karen; Diamond, Beverly

doi:10.1090/S0002-9939-96-03690-8

Self-similarity in inverse limit spaces of the tent family
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by Marcy Barge, Karen Brucks and Beverly Diamond PDF

Proc. Amer. Math. Soc. 124 (1996), 3563-3570 Request permission

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

Marcy Barge and Sarah Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8 (1995), no. 1, 29–42. MR 1313141, DOI 10.1088/0951-7715/8/1/003
Marcy Barge and Joe Martin, Endpoints of inverse limit spaces and dynamics, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 165–182. MR 1326840
Morton Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478–483. MR 115157, DOI 10.1090/S0002-9939-1960-0115157-4
K. M. Brucks and B. Diamond, Monotonicity of auto-expansions, Phys. D 51 (1991), no. 1-3, 39–42. Nonlinear science: the next decade (Los Alamos, NM, 1990). MR 1128801, DOI 10.1016/0167-2789(91)90220-4
K. M. Brucks, B. Diamond, M. V. Otero-Espinar, and C. Tresser, Dense orbits of critical points for the tent map, Continuum theory and dynamical systems (Arcata, CA, 1989) Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 57–61. MR 1112803, DOI 10.1090/conm/117/1112803
Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR 613981
Ethan M. Coven, Ittai Kan, and James A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), no. 1, 227–241. MR 946440, DOI 10.1090/S0002-9947-1988-0946440-2
S. Holte, Inverse limits of Markov interval maps, preprint.
C. Robinson, Dynamical Systems, CRC, Boca Raton, 1995.
R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473–487. MR 217808, DOI 10.1016/0040-9383(67)90005-5