# American Institute of Mathematical Sciences

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

## On isotopy and unimodal inverse limit spaces

 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 $\sigma^R : K_s \to K_s$.
Citation: Henk Bruin, Sonja Štimac. On isotopy and unimodal inverse limit spaces. Discrete and 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. [2] M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, preprint, 2009, arXiv:0912.4645. [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. [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. [5] K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246. [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. [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. [8] H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proceedings, 31 (2007), 37-50. [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. [10] B. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268. doi: 10.4064/fm182-3-4. [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. [12] S. Štimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl., 154 (2007), 2265-2281.

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. [2] M. Barge, H. Bruin and S. Štimac, The Ingram conjecture, preprint, 2009, arXiv:0912.4645. [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. [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. [5] K. Brucks and H. Bruin, Subcontinua of inverse limit spaces of unimodal maps, Fund. Math., 160 (1999), 219-246. [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. [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. [8] H. Bruin, Subcontinua of Fibonacci-like unimodal inverse limit spaces, Topology Proceedings, 31 (2007), 37-50. [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. [10] B. Raines, Inhomogeneities in non-hyperbolic one-dimensional invariant sets, Fund. Math., 182 (2004), 241-268. doi: 10.4064/fm182-3-4. [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. [12] S. Štimac, A classification of inverse limit spaces of tent maps with finite critical orbit, Topology Appl., 154 (2007), 2265-2281.
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