August  2013, 33(8): 3277-3287. doi: 10.3934/dcds.2013.33.3277

Toeplitz kneading sequences and adding machines

1. 

Department of Mathematics and Statistics, University of West Florida, 11000 University Parkway, Pensacola, FL 32514, United States

Received  May 2012 Revised  November 2012 Published  January 2013

In this paper we provide a characterization for a shift maximal sequence of 1's and 0's to be the kneading sequence for a unimodal map $f$ with $f|_{\omega(c)}$ topologically conjugate to an adding machine, where $c$ is the turning point of $f$. We show that the unimodal map $f$ has an embedded adding machine if and only if $\mathcal{K}(f)$ is a one-sided, non-periodic Toeplitz sequence with the finite time containment property. We then show the existence of unimodal maps with Toeplitz kneading sequences that do not have the finite time containment property.
Citation: Lori Alvin. Toeplitz kneading sequences and adding machines. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3277-3287. doi: 10.3934/dcds.2013.33.3277
References:
[1]

L. Alvin, The strange star product, J. Difference Eq. and Appl., 18 (2012), 657-674. doi: 10.1080/10236198.2011.608066.

[2]

L. Alvin and K. Brucks, Adding machines, endpoints, and inverse limit spaces, Fund. Math., 209 (2010), 81-93. doi: 10.4064/fm209-1-6.

[3]

W. A. Beyer, R. D. Mauldin and P. R. Stein, Shift maximal sequences in function iteration: Existence, uniqueness, and multiplicity, J. Math. Anal. Appl., 115 (1986), 305-362 doi: 10.1016/0022-247X(86)90001-6.

[4]

L. Block and J. Keesling, A characterization of adding machines maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006.

[5]

L. Block, J. Keesling and M. Misiurewicz, Strange adding machines, Ergod. Th. & Dynam. Sys., 26 (2006), 673-682 . doi: 10.1017/S0143385705000635.

[6]

K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," Cambridge University Press, 2004.

[7]

P. Collet and J-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhauser, 1980.

[8]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Commun. Math. Phys., 127 (1990), 319-337.

[9]

L. Jones, Kneading sequences of strange adding machines, Topology Appl., 156 (2009), 2735-2746. doi: 10.1016/j.topol.2008.11.018.

[10]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Springer Verlag, 1993.

[11]

S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107. doi: 10.1007/BF00534085.

show all references

References:
[1]

L. Alvin, The strange star product, J. Difference Eq. and Appl., 18 (2012), 657-674. doi: 10.1080/10236198.2011.608066.

[2]

L. Alvin and K. Brucks, Adding machines, endpoints, and inverse limit spaces, Fund. Math., 209 (2010), 81-93. doi: 10.4064/fm209-1-6.

[3]

W. A. Beyer, R. D. Mauldin and P. R. Stein, Shift maximal sequences in function iteration: Existence, uniqueness, and multiplicity, J. Math. Anal. Appl., 115 (1986), 305-362 doi: 10.1016/0022-247X(86)90001-6.

[4]

L. Block and J. Keesling, A characterization of adding machines maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006.

[5]

L. Block, J. Keesling and M. Misiurewicz, Strange adding machines, Ergod. Th. & Dynam. Sys., 26 (2006), 673-682 . doi: 10.1017/S0143385705000635.

[6]

K. M. Brucks and H. Bruin, "Topics From One-Dimensional Dynamics," Cambridge University Press, 2004.

[7]

P. Collet and J-P. Eckmann, "Iterated Maps on the Interval as Dynamical Systems," Birkhauser, 1980.

[8]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Commun. Math. Phys., 127 (1990), 319-337.

[9]

L. Jones, Kneading sequences of strange adding machines, Topology Appl., 156 (2009), 2735-2746. doi: 10.1016/j.topol.2008.11.018.

[10]

W. de Melo and S. van Strien, "One-Dimensional Dynamics," Springer Verlag, 1993.

[11]

S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete, 67 (1984), 95-107. doi: 10.1007/BF00534085.

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