# Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms

Colloquium Mathematicae (1995)

- Volume: 68, Issue: 2, page 291-296
- ISSN: 0010-1354

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topHolte, Sarah. "Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms." Colloquium Mathematicae 68.2 (1995): 291-296. <http://eudml.org/doc/210313>.

@article{Holte1995,

abstract = {In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism $\widehat\{f\}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism $F:ℝ^2 → ℝ^2$ so that F restricted to its full attracting set, $⋂_\{k ≥ 0\} F^k(ℝ^2)$, is topologically conjugate to $\widehat\{f\}:(I,f) → (I,f)$. In this situation, we say that the inverse limit space, (I,f), can be embedded as the full attracting set of F.},

author = {Holte, Sarah},

journal = {Colloquium Mathematicae},

keywords = {inverse limit; interval map; plane diffeomorphism},

language = {eng},

number = {2},

pages = {291-296},

title = {Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms},

url = {http://eudml.org/doc/210313},

volume = {68},

year = {1995},

}

TY - JOUR

AU - Holte, Sarah

TI - Embedding inverse limits of nearly Markov interval maps as attracting sets of planar diffeomorphisms

JO - Colloquium Mathematicae

PY - 1995

VL - 68

IS - 2

SP - 291

EP - 296

AB - In this paper we address the following question due to Marcy Barge: For what f:I → I is it the case that the inverse limit of I with single bonding map f can be embedded in the plane so that the shift homeomorphism $\widehat{f}$ extends to a diffeomorphism ([BB, Problem 1.5], [BK, Problem 3])? This question could also be phrased as follows: Given a map f:I → I, find a diffeomorphism $F:ℝ^2 → ℝ^2$ so that F restricted to its full attracting set, $⋂_{k ≥ 0} F^k(ℝ^2)$, is topologically conjugate to $\widehat{f}:(I,f) → (I,f)$. In this situation, we say that the inverse limit space, (I,f), can be embedded as the full attracting set of F.

LA - eng

KW - inverse limit; interval map; plane diffeomorphism

UR - http://eudml.org/doc/210313

ER -

## References

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- [BB] M. Barge and M. Brown, Problems in dynamics on continua, in: Continuum Theory and Dynamical Systems, M. Brown (ed.), Amer. Math. Soc., Providence, R.I., 1991, 177-182.
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- [BM] M. Barge and J. Martin, The construction of global attractors, Proc. Amer. Math. Soc. 110 (1990), 523-525. Zbl0714.58036
- [Bl] L. Block, Diffeomorphisms obtained from endomorphisms, Trans. Amer. Math. Soc. 214, 403-413. Zbl0293.57016
- [H] S. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-305. Zbl0723.58034
- [M] M. Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985), 23-40. Zbl0587.58032
- [Sc] R. Schori, Chaos: An introduction to some topological aspects, in: Continuum Theory and Dynamical Systems, M. Brown (ed.), Amer. Math. Soc., Providence, R.I., 1991, 149-161.
- [Sz] W. Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989), 1-23. Zbl0708.58012
- [W] R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473-487. Zbl0159.53702

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