We discuss the inverse limit spaces of unimodal interval maps as topological spaces. Based on the combinatorial properties of the unimodal maps, properties of the subcontinua of the inverse limit spaces are studied. Among other results, we give combinatorial conditions for an inverse limit space to have only arc+ray subcontinua as proper (non-trivial) subcontinua. Also, maps are constructed whose inverse limit spaces have the inverse limit spaces of a prescribed set of periodic unimodal maps as subcontinua.
@article{bwmeta1.element.bwnjournal-article-fmv160i3p219bwm,
author = {Karen Brucks and Henk Bruin},
title = {Subcontinua of inverse limit spaces of unimodal maps},
journal = {Fundamenta Mathematicae},
volume = {159},
year = {1999},
pages = {219-246},
zbl = {0953.54032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p219bwm}
}
Brucks, Karen; Bruin, Henk. Subcontinua of inverse limit spaces of unimodal maps. Fundamenta Mathematicae, Tome 159 (1999) pp. 219-246. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv160i3p219bwm/
[00000] [1] J. M. Aarts and R. J. Fokkink, The classification of solenoids, Proc. Amer. Math. Soc. 111 (1991), 1161-1163. | Zbl 0768.54026
[00001] [2] M. Barge, Horseshoe maps and inverse limits, Pacific J. Math. 121 (1986), 29-39. | Zbl 0601.58049
[00002] [3] M. Barge, K. Brucks and B. Diamond, Self-similarity in inverse limit spaces of the tent family, Proc. Amer. Math. Soc. 124 (1996), 3563-3570. | Zbl 0917.54041
[00003] [4] M. Barge and B. Diamond, Homeomorphisms of inverse limit spaces of one-dimensional maps, Fund. Math. 146 (1995), 171-187.
[00004] [5] M. Barge and B. Diamond, Inverse limit spaces of infinitely renormalizable maps, Topology Appl. 83 (1998), 103-108. | Zbl 0967.54031
[00005] [6] M. Barge and B. Diamond, Subcontinua of the closure of the unstable manifold at a homoclinic tangency, Ergodic Theory Dynam. Systems 19 (1999), 1-19.
[00006] [7] M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8 (1995), 29-42.
[00007] [8] M. Barge and J. Martin, Endpoints of inverse limit spaces and dynamics, in: Continua with the Houston Problem Book (Cincinnati, OH, 1994), Lecture Notes in Pure and Appl. Math. 170, Dekker, New York, 1995, 165-182. | Zbl 0826.58023
[00008] [9] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. | Zbl 0113.37705
[00009] [10] K. Brucks, B. Diamond, M. V. Otero-Espinar and C. Tresser, Dense orbits of critical points for the tent map, in: Contemp. Math. 117, Amer. Math. Soc., Providence, RI, 1991, 57-61. | Zbl 0746.34029
[00010] [11] H. Bruin, Invariant measures of interval maps, Ph.D. thesis, Delft, 1994. | Zbl 0812.58052
[00011] [12] H. Bruin, Combinatorics of the kneading map, Internat J. Bifur. and Chaos Appl. Sci. Engrg. 5 (1995), 1339-1349. | Zbl 0886.58023
[00012] [13] H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., to appear. | Zbl 0954.54019
[00013] [14] H. Bruin, Inverse limit spaces of post-critically finite tent maps, preprint, 1998. | Zbl 0973.37011
[00014] [15] D W. Dębski, On topological types of the simplest indecomposable continua, Colloq. Math. 49 (1985), 203-211. | Zbl 0591.54026
[00015] [16] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980), 117-141. | Zbl 0433.54009
[00016] [17] F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319-337. | Zbl 0702.58034
[00017] [18] S. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-305. | Zbl 0723.58034
[00018] [19] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993. | Zbl 0791.58003
[00019] [20] J. Mioduszewski, Mappings of inverse limits, Colloq. Math. 10 (1963), 39-44. | Zbl 0118.18205
[00020] [21] S. Nadler, Continuum Theory, Dekker, New York, 1992.
[00021] [22] R. C. Swanson and H. W. Volkmer, Invariants of weak equivalence in primitive matrices, preprint, 1998. | Zbl 0984.37019
[00022] [23] W W. T. Watkins, Homeomorphic classification of certain inverse limit spaces with open bonding maps, Pacific J. Math. 103 (1982), 589-601. | Zbl 0451.54027
[00023] [24] R. F. Williams, One-dimensional nonwandering sets, Topology 6 (1967), 473-487. | Zbl 0159.53702