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

Toeplitz kneading sequences and adding machines

Lori Alvin 1,

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.
Keywords: Adding machines, Toeplitz sequences, kneading sequences, unimodal maps..
Mathematics Subject Classification: Primary: 54H20, 37B10; Secondary: 37E0.
Citation: Lori Alvin. Toeplitz kneading sequences and adding machines. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1080/10236198.2011.608066.  Google Scholar

[2]

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

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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.  doi: 10.1016/0022-247X(86)90001-6.  Google Scholar

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

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

show all references

References:
[1]

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

[2]

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

[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.  doi: 10.1016/0022-247X(86)90001-6.  Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

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