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 & 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

[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

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

[1]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313

[2]

Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5951-5970. doi: 10.3934/dcds.2016061

[3]

Nuno Franco, Luís Silva. Genus and braid index associated to sequences of renormalizable Lorenz maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 565-586. doi: 10.3934/dcds.2012.32.565

[4]

Mrinal Kanti Roychowdhury, Daniel J. Rudolph. Nearly continuous Kakutani equivalence of adding machines. Journal of Modern Dynamics, 2009, 3 (1) : 103-119. doi: 10.3934/jmd.2009.3.103

[5]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010

[6]

Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379

[7]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933

[8]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617

[9]

Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199

[10]

Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81

[11]

A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587

[12]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

[13]

Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178

[14]

Yu Zheng, Li Peng, Teturo Kamae. Characterization of noncorrelated pattern sequences and correlation dimensions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5085-5103. doi: 10.3934/dcds.2018223

[15]

Jörg Härterich, Matthias Wolfrum. Describing a class of global attractors via symbol sequences. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 531-554. doi: 10.3934/dcds.2005.12.531

[16]

Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229

[17]

Jiarong Peng, Xiangyong Zeng, Zhimin Sun. Finite length sequences with large nonlinear complexity. Advances in Mathematics of Communications, 2018, 12 (1) : 215-230. doi: 10.3934/amc.2018015

[18]

Xiaoni Du, Chenhuang Wu, Wanyin Wei. An extension of binary threshold sequences from Fermat quotients. Advances in Mathematics of Communications, 2016, 10 (4) : 743-752. doi: 10.3934/amc.2016038

[19]

Alexei Pokrovskii, Oleg Rasskazov. Structure of index sequences for mappings with an asymptotic derivative. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 653-670. doi: 10.3934/dcds.2007.17.653

[20]

Santos González, Llorenç Huguet, Consuelo Martínez, Hugo Villafañe. Discrete logarithm like problems and linear recurring sequences. Advances in Mathematics of Communications, 2013, 7 (2) : 187-195. doi: 10.3934/amc.2013.7.187

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]