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

Sarah Holte

Colloquium Mathematicae (1995)

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

Abstract

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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 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 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.

How to cite

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Holte, 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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  1. [Ba] M. Barge, A method for construction of attractors, Ergodic Theory Dynamical Systems 8 (1988), 331-349. 
  2. [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. 
  3. [BK] M. Barge and J. Kennedy, Continuum theory and topological dynamics, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, Amsterdam, 1990, 633-644. 
  4. [BM] M. Barge and J. Martin, The construction of global attractors, Proc. Amer. Math. Soc. 110 (1990), 523-525. Zbl0714.58036
  5. [Bl] L. Block, Diffeomorphisms obtained from endomorphisms, Trans. Amer. Math. Soc. 214, 403-413. Zbl0293.57016
  6. [H] S. Holte, Generalized horseshoe maps and inverse limits, Pacific J. Math. 156 (1992), 297-305. Zbl0723.58034
  7. [M] M. Misiurewicz, Embedding inverse limits of interval maps as attractors, Fund. Math. 125 (1985), 23-40. Zbl0587.58032
  8. [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. 
  9. [Sz] W. Szczechla, Inverse limits of certain interval mappings as attractors in two dimensions, Fund. Math. 133 (1989), 1-23. Zbl0708.58012
  10. [W] R. F. Williams, One-dimensional non-wandering sets, Topology 6 (1967), 473-487. Zbl0159.53702

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