American Institute of Mathematical Sciences

Advanced
May  2003, 9(3): 549-558. doi: 10.3934/dcds.2003.9.549

A twisted tensor product on symbolic dynamical systems and the Ashley's problem

H. M. Hastings 1, , S. Silberger 1, , M. T. Weiss 1, and  Y. Wu 1,

1. 

Department of Mathematics, Hofstra University, Hempstead, NY 11550, United States, United States, United States, United States

Received  February 2002 Revised  November 2002 Published  February 2003

We define the notion of fiber bundle via a twisted tensor product on the transition matrices. We define the notion of topological conjugacy and shift equivalence in this bundle context and show that topological conjugacy implies shift equivalence. We show that the "Ashley system" $\Sigma_A$ fits into our fiber bundle context. We introduce another system $\Sigma_W$, topologically conjugate to the full $2-$shift, which has the same base space and fiber as the Ashley system, but is constructed with a different twisting. We show that $\Sigma_A$ and $\Sigma_W$ are shift equivalent but not bundle isomorphic.
Keywords: Symbolic dynamics, shift equivalence, topological conjugacy, Ashley’s problem..
Mathematics Subject Classification: Primary: 37B10, 37C1.
Citation: H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[1]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[2]

Nicola Soave, Susanna Terracini. Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3825-3829. doi: 10.3934/dcds.2013.33.3825
[3]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541
[4]

Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161
[5]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[6]

Adriano Da Silva, Alexandre J. Santana, Simão N. Stelmastchuk. Topological conjugacy of linear systems on Lie groups. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3411-3421. doi: 10.3934/dcds.2017144
[7]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[8]

Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557
[9]

A. Yu. Ol'shanskii and M. V. Sapir. The conjugacy problem for groups, and Higman embeddings. Electronic Research Announcements, 2003, 9: 40-50.

[10]

David Burguet, Ruxi Shi. Zero-dimensional and symbolic extensions of topological flows. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1105-1126. doi: 10.3934/dcds.2021148
[11]

Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247
[12]

James Kingsbery, Alex Levin, Anatoly Preygel, Cesar E. Silva. Dynamics of the $p$-adic shift and applications. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 209-218. doi: 10.3934/dcds.2011.30.209
[13]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621
[14]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43
[15]

Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301
[16]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
[17]

Fritz Colonius, Alexandre J. Santana. Topological conjugacy for affine-linear flows and control systems. Communications on Pure and Applied Analysis, 2011, 10 (3) : 847-857. doi: 10.3934/cpaa.2011.10.847
[18]

Ming-Chia Li, Ming-Jiea Lyu. Topological conjugacy for Lipschitz perturbations of non-autonomous systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5011-5024. doi: 10.3934/dcds.2016017
[19]

Álvaro Castañeda, Gonzalo Robledo. Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2287-2304. doi: 10.3934/dcds.2018094
[20]

Wei Lin, Jianhong Wu, Guanrong Chen. Generalized snap-back repeller and semi-conjugacy to shift operators of piecewise continuous transformations. Discrete and Continuous Dynamical Systems, 2007, 19 (1) : 103-119. doi: 10.3934/dcds.2007.19.103

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy