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April  2012, 32(4): 1245-1253. doi: 10.3934/dcds.2012.32.1245

On isotopy and unimodal inverse limit spaces

Henk Bruin 1, and  Sonja Štimac 2,

1. 

Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom
2. 

Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia

Received  October 2010 Revised  July 2011 Published  October 2011

We prove that every self-homeomorphism $h : K_s \to K_s$ on the inverse limit space $K_s$ of tent map $T_s$ with slope $s \in (\sqrt 2, 2]$ is isotopic to a power of the shift-homeomorphism $\sigma^R : K_s \to K_s$.
Keywords: inverse limit space., Isotopy, tent map.
Mathematics Subject Classification: Primary: 54H20; Secondary: 37B45, 37E0.
Citation: Henk Bruin, Sonja Štimac. On isotopy and unimodal inverse limit spaces. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1245-1253. doi: 10.3934/dcds.2012.32.1245
References:
[1]

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. doi: 10.1090/S0002-9939-96-03690-8.  Google Scholar

[2]

M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, preprint, 2009, arXiv:0912.4645. Google Scholar

[3]

L. Block, S. Jakimovik, J. Keesling and L. Kailhofer, On the classification of inverse limits of tent maps, Fund. Math., 187 (2005), 171-192. doi: 10.4064/fm187-2-5.  Google Scholar

[4]

L. Block, J. Keesling, B. Raines and S. Štimac, Homeomorphisms of unimodal inverse limit spaces with non-recurrent critical point, Topology Appl., 156 (2009), 2417-2425. doi: 10.1016/j.topol.2009.06.006.  Google Scholar

[5]

K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246.  Google Scholar

[6]

K. Brucks and B. Diamond, A symbolic representation of inverse limit spaces for a class of unimodal maps, in "Continua" (Cincinnati, OH, 1994), 207-226, Lect. Notes in Pure and Appl. Math. 170, Dekker, New York, 1995.  Google Scholar

[7]

K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergod. Th. and Dyn. Sys., 16 (1996), 1173-1183. doi: 10.1017/S0143385700009962.  Google Scholar

[8]

H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proceedings, 31 (2007), 37-50.  Google Scholar

[9]

L. Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math., 177 (2003), 95-120. doi: 10.4064/fm177-2-1.  Google Scholar

[10]

B. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268. doi: 10.4064/fm182-3-4.  Google Scholar

[11]

B. Raines and S. Štimac, A classification of inverse limit spaces of tent maps with nonrecurrent critical point, Algebraic and Geometric Topology, 9 (2009), 1049-1088. doi: 10.2140/agt.2009.9.1049.  Google Scholar

[12]

S. Štimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl., 154 (2007), 2265-2281.  Google Scholar

show all references

References:
[1]

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. doi: 10.1090/S0002-9939-96-03690-8.  Google Scholar

[2]

M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, preprint, 2009, arXiv:0912.4645. Google Scholar

[3]

L. Block, S. Jakimovik, J. Keesling and L. Kailhofer, On the classification of inverse limits of tent maps, Fund. Math., 187 (2005), 171-192. doi: 10.4064/fm187-2-5.  Google Scholar

[4]

L. Block, J. Keesling, B. Raines and S. Štimac, Homeomorphisms of unimodal inverse limit spaces with non-recurrent critical point, Topology Appl., 156 (2009), 2417-2425. doi: 10.1016/j.topol.2009.06.006.  Google Scholar

[5]

K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246.  Google Scholar

[6]

K. Brucks and B. Diamond, A symbolic representation of inverse limit spaces for a class of unimodal maps, in "Continua" (Cincinnati, OH, 1994), 207-226, Lect. Notes in Pure and Appl. Math. 170, Dekker, New York, 1995.  Google Scholar

[7]

K. Brucks and M. Misiurewicz, The trajectory of the turning point is dense for almost all tent maps, Ergod. Th. and Dyn. Sys., 16 (1996), 1173-1183. doi: 10.1017/S0143385700009962.  Google Scholar

[8]

H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proceedings, 31 (2007), 37-50.  Google Scholar

[9]

L. Kailhofer, A classification of inverse limit spaces of tent maps with periodic critical points, Fund. Math., 177 (2003), 95-120. doi: 10.4064/fm177-2-1.  Google Scholar

[10]

B. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268. doi: 10.4064/fm182-3-4.  Google Scholar

[11]

B. Raines and S. Štimac, A classification of inverse limit spaces of tent maps with nonrecurrent critical point, Algebraic and Geometric Topology, 9 (2009), 1049-1088. doi: 10.2140/agt.2009.9.1049.  Google Scholar

[12]

S. Štimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl., 154 (2007), 2265-2281.  Google Scholar

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