# Subcontinua of inverse limit spaces of unimodal maps

Fundamenta Mathematicae (1999)

- Volume: 160, Issue: 3, page 219-246
- ISSN: 0016-2736

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topBrucks, Karen, and Bruin, Henk. "Subcontinua of inverse limit spaces of unimodal maps." Fundamenta Mathematicae 160.3 (1999): 219-246. <http://eudml.org/doc/212390>.

@article{Brucks1999,

abstract = {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.},

author = {Brucks, Karen, Bruin, Henk},

journal = {Fundamenta Mathematicae},

keywords = {unimodal maps; inverse limit},

language = {eng},

number = {3},

pages = {219-246},

title = {Subcontinua of inverse limit spaces of unimodal maps},

url = {http://eudml.org/doc/212390},

volume = {160},

year = {1999},

}

TY - JOUR

AU - Brucks, Karen

AU - Bruin, Henk

TI - Subcontinua of inverse limit spaces of unimodal maps

JO - Fundamenta Mathematicae

PY - 1999

VL - 160

IS - 3

SP - 219

EP - 246

AB - 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.

LA - eng

KW - unimodal maps; inverse limit

UR - http://eudml.org/doc/212390

ER -

## References

top- [1] J. M. Aarts and R. J. Fokkink, The classification of solenoids, Proc. Amer. Math. Soc. 111 (1991), 1161-1163. Zbl0768.54026
- [2] M. Barge, Horseshoe maps and inverse limits, Pacific J. Math. 121 (1986), 29-39. Zbl0601.58049
- [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. Zbl0917.54041
- [4] M. Barge and B. Diamond, Homeomorphisms of inverse limit spaces of one-dimensional maps, Fund. Math. 146 (1995), 171-187.
- [5] M. Barge and B. Diamond, Inverse limit spaces of infinitely renormalizable maps, Topology Appl. 83 (1998), 103-108. Zbl0967.54031
- [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.
- [7] M. Barge and S. Holte, Nearly one-dimensional Hénon attractors and inverse limits, Nonlinearity 8 (1995), 29-42.
- [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. Zbl0826.58023
- [9] M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. Zbl0113.37705
- [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. Zbl0746.34029
- [11] H. Bruin, Invariant measures of interval maps, Ph.D. thesis, Delft, 1994. Zbl0812.58052
- [12] H. Bruin, Combinatorics of the kneading map, Internat J. Bifur. and Chaos Appl. Sci. Engrg. 5 (1995), 1339-1349. Zbl0886.58023
- [13] H. Bruin, Planar embeddings of inverse limit spaces of unimodal maps, Topology Appl., to appear. Zbl0954.54019
- [14] H. Bruin, Inverse limit spaces of post-critically finite tent maps, preprint, 1998. Zbl0973.37011
- [15] D W. Dębski, On topological types of the simplest indecomposable continua, Colloq. Math. 49 (1985), 203-211. Zbl0591.54026
- [16] F. Hofbauer, The topological entropy of the transformation x ↦ ax(1-x), Monatsh. Math. 90 (1980), 117-141. Zbl0433.54009
- [17] F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys. 127 (1990), 319-337. Zbl0702.58034
- [18] S. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-305. Zbl0723.58034
- [19] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993. Zbl0791.58003
- [20] J. Mioduszewski, Mappings of inverse limits, Colloq. Math. 10 (1963), 39-44. Zbl0118.18205
- [21] S. Nadler, Continuum Theory, Dekker, New York, 1992.
- [22] R. C. Swanson and H. W. Volkmer, Invariants of weak equivalence in primitive matrices, preprint, 1998. Zbl0984.37019
- [23] W W. T. Watkins, Homeomorphic classification of certain inverse limit spaces with open bonding maps, Pacific J. Math. 103 (1982), 589-601. Zbl0451.54027
- [24] R. F. Williams, One-dimensional nonwandering sets, Topology 6 (1967), 473-487. Zbl0159.53702

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