American Institute of Mathematical Sciences

Advanced
October  2015, 35(10): 4743-4764. doi: 10.3934/dcds.2015.35.4743

Transitive sofic spacing shifts

John Banks 1, , Piotr Oprocha 2, and  Brett Stanley 3,

1. 

Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Australia
2. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków
3. 

Department of Mathematics and Statistics, La Trobe University, Bundoora 3086, Australia

Received  May 2014 Revised  February 2015 Published  April 2015

Spacing shifts were introduced by Lau and Zame in the 1970's to provide accessible examples of maps that are weakly mixing but not mixing. In previous papers by the authors and others, it has been observed that the problem of describing when spacing shifts are topologically transitive appears to be quite difficult in general. In the present paper, we give a characterization of sofic spacing shifts and begin to investigate which sofic spacing shifts are topologically transitive. We show that the canonical graph presentation of such a shift has a rather simple form, for which we introduce the terminology hereditary bunched cycle and discuss the apparently difficult problem of determining which hereditary bunched cycles actually present spacing shifts.
Keywords: Symbolic dynamics, sofic shifts, topological transitivity, spacing shifts..
Mathematics Subject Classification: Primary: 37B10; Secondary: 37B20, 05C38, 68Q4.
Citation: John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743
References:
[1]

D. Ahmadi and M. Dabbaghian, Characterization of spacing shifts with positive topological entropy,, Acta Math. Univ. Comenian. (N.S.), 81 (2012), 221.   Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances,, North-Holland Mathematical Library, (1994).   Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps,, Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[4]

J. Banks, T. T. D. Nguyen, P. Oprocha, B. Stanley and B. Trotta, Dynamics of spacing shifts,, Discrete Contin. Dyn. Syst. 33 (2013), 33 (2013), 4207.  doi: 10.3934/dcds.2013.33.4207.  Google Scholar

[5]

D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition,, John Wiley & Sons, (2004).   Google Scholar

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).   Google Scholar

[7]

S. Ginsburg, Algebraic and automata-theoretic properties of formal languages,, North-Holland/American Elsevier, (1975).   Google Scholar

[8]

M. Harrison, Introduction to Formal Language Theory,, Addison-Wesley, (1978).   Google Scholar

[9]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts,, Discrete Contin. Dyn. Syst, 33 (2013), 2451.  doi: 10.3934/dcds.2013.33.2451.  Google Scholar

[10]

K. Lau and A. Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (1973), 307.  doi: 10.1007/BF01740722.  Google Scholar

[11]

D. Lind and B. Marcus, Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511626302.  Google Scholar

[12]

M. Lothaire, Algebraic Combinatorics on Words,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511566097.  Google Scholar

show all references

References:
[1]

D. Ahmadi and M. Dabbaghian, Characterization of spacing shifts with positive topological entropy,, Acta Math. Univ. Comenian. (N.S.), 81 (2012), 221.   Google Scholar

[2]

N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems. Recent Advances,, North-Holland Mathematical Library, (1994).   Google Scholar

[3]

J. Banks, Regular Periodic Decompositions for Topologically Transitive maps,, Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[4]

J. Banks, T. T. D. Nguyen, P. Oprocha, B. Stanley and B. Trotta, Dynamics of spacing shifts,, Discrete Contin. Dyn. Syst. 33 (2013), 33 (2013), 4207.  doi: 10.3934/dcds.2013.33.4207.  Google Scholar

[5]

D. S. Dummit and R. M. Foote, Abstract Algebra, Third edition,, John Wiley & Sons, (2004).   Google Scholar

[6]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).   Google Scholar

[7]

S. Ginsburg, Algebraic and automata-theoretic properties of formal languages,, North-Holland/American Elsevier, (1975).   Google Scholar

[8]

M. Harrison, Introduction to Formal Language Theory,, Addison-Wesley, (1978).   Google Scholar

[9]

D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts,, Discrete Contin. Dyn. Syst, 33 (2013), 2451.  doi: 10.3934/dcds.2013.33.2451.  Google Scholar

[10]

K. Lau and A. Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (1973), 307.  doi: 10.1007/BF01740722.  Google Scholar

[11]

D. Lind and B. Marcus, Introduction to Symbolic Dynamics and Coding,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511626302.  Google Scholar

[12]

M. Lothaire, Algebraic Combinatorics on Words,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511566097.  Google Scholar

[1]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020278
[2]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350
[3]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127
[4]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[5]

Jong-Shenq Guo, Ken-Ichi Nakamura, Toshiko Ogiwara, Chang-Hong Wu. The sign of traveling wave speed in bistable dynamics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3451-3466. doi: 10.3934/dcds.2020047
[6]

Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020365
[7]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362
[8]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406
[9]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[10]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[11]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[12]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[13]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426
[14]

Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085
[15]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360
[16]

Linfeng Mei, Feng-Bin Wang. Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020359
[17]

Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020363
[18]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288
[19]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263
[20]

Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences