September  2013, 33(9): 4207-4232. doi: 10.3934/dcds.2013.33.4207

Dynamics of spacing shifts

1. 

La Trobe University, Bundoora 3086, Australia, Australia, Australia

2. 

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

3. 

AGL Energy, 120 Spencer St, Melbourne VIC 3000, Australia

Received  November 2010 Revised  February 2011 Published  March 2013

Spacing subshifts were introduced by Lau and Zame in 1973 to provide accessible examples of maps that are (topologically) weakly mixing but not mixing. Although they show a rich variety of dynamical characteristics, they have received little subsequent attention in the dynamical systems literature. This paper is a systematic study of their dynamical properties and shows that they may be used to provide examples of dynamical systems with a huge range of interesting dynamical behaviors. In a later paper we propose to consider in more detail the case when spacing subshifts are also sofic and transitive.
Citation: John Banks, Thi T. D. Nguyen, Piotr Oprocha, Brett Stanley, Belinda Trotta. Dynamics of spacing shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4207-4232. doi: 10.3934/dcds.2013.33.4207
References:
[1]

E. Akin and S. Kolyada, Li-Yorke sensitivity,, Nonlinearity, 16 (2003), 1421.  doi: 10.1088/0951-7715/16/4/313.  Google Scholar

[2]

E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems,, in, 356 (2004), 21.  doi: 10.1090/conm/356/06496.  Google Scholar

[3]

L. Alseda, M. del Rio and J. Rodriguez, Transitivity and dense periodicity for graph maps,, J. Diff. Eq. Applications, 9 (2003), 577.  doi: 10.1080/1023619021000040515.  Google Scholar

[4]

F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets,, Colloq. Math., 110 (2008), 293.  doi: 10.4064/cm110-2-3.  Google Scholar

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[6]

J. Banks and B. Trotta, Weak mixing implies mixing for maps on topological graphs,, J. Diff. Eq. Applications, 11 (2005), 1071.  doi: 10.1080/1023619050029557.  Google Scholar

[7]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51.  doi: 10.1515/crll.2002.053.  Google Scholar

[8]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309.   Google Scholar

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401.   Google Scholar

[10]

H. Furstenberg, Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1.   Google Scholar

[11]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", M. B. Porter Lectures, (1981).   Google Scholar

[12]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos,, Topology Appl., 117 (2002), 259.  doi: 10.1016/S0166-8641(01)00025-6.  Google Scholar

[13]

A. Iwanik, Independence and scrambled sets for chaotic mappings,, in, (1991), 372.   Google Scholar

[14]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics,, Math. Ann., 338 (2007), 869.  doi: 10.1007/s00208-007-0097-z.  Google Scholar

[15]

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

[16]

G. Liao, Z. Chu and Q. Fan, Relations between mixing and distributional chaos,, Chaos Solitons Fractals, 41 (2009), 1994.  doi: 10.1016/j.chaos.2008.08.003.  Google Scholar

[17]

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

[18]

Edwin E. Moise, "Geometric Topology in Dimensions $2$ and $3$,", Graduate Texts in Mathematics, (1977).   Google Scholar

[19]

P. Oprocha, Relations between distributional and Devaney chaos,, Chaos, 16 (2006).  doi: 10.1063/1.2225513.  Google Scholar

[20]

B. Schweizer and J. Smítal, Measures of chaos and spectral decomposition of dynamical systems on the interval,, Trans. Amer. Math. Soc., 344 (1994), 737.  doi: 10.2307/2154504.  Google Scholar

[21]

J. C. Xiong, A chaotic map with topological entropy,, Acta Math. Sci. (English Ed.), 6 (1986), 439.   Google Scholar

show all references

References:
[1]

E. Akin and S. Kolyada, Li-Yorke sensitivity,, Nonlinearity, 16 (2003), 1421.  doi: 10.1088/0951-7715/16/4/313.  Google Scholar

[2]

E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems,, in, 356 (2004), 21.  doi: 10.1090/conm/356/06496.  Google Scholar

[3]

L. Alseda, M. del Rio and J. Rodriguez, Transitivity and dense periodicity for graph maps,, J. Diff. Eq. Applications, 9 (2003), 577.  doi: 10.1080/1023619021000040515.  Google Scholar

[4]

F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets,, Colloq. Math., 110 (2008), 293.  doi: 10.4064/cm110-2-3.  Google Scholar

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505.  doi: 10.1017/S0143385797069885.  Google Scholar

[6]

J. Banks and B. Trotta, Weak mixing implies mixing for maps on topological graphs,, J. Diff. Eq. Applications, 11 (2005), 1071.  doi: 10.1080/1023619050029557.  Google Scholar

[7]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51.  doi: 10.1515/crll.2002.053.  Google Scholar

[8]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309.   Google Scholar

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401.   Google Scholar

[10]

H. Furstenberg, Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1.   Google Scholar

[11]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory,", M. B. Porter Lectures, (1981).   Google Scholar

[12]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos,, Topology Appl., 117 (2002), 259.  doi: 10.1016/S0166-8641(01)00025-6.  Google Scholar

[13]

A. Iwanik, Independence and scrambled sets for chaotic mappings,, in, (1991), 372.   Google Scholar

[14]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics,, Math. Ann., 338 (2007), 869.  doi: 10.1007/s00208-007-0097-z.  Google Scholar

[15]

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

[16]

G. Liao, Z. Chu and Q. Fan, Relations between mixing and distributional chaos,, Chaos Solitons Fractals, 41 (2009), 1994.  doi: 10.1016/j.chaos.2008.08.003.  Google Scholar

[17]

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

[18]

Edwin E. Moise, "Geometric Topology in Dimensions $2$ and $3$,", Graduate Texts in Mathematics, (1977).   Google Scholar

[19]

P. Oprocha, Relations between distributional and Devaney chaos,, Chaos, 16 (2006).  doi: 10.1063/1.2225513.  Google Scholar

[20]

B. Schweizer and J. Smítal, Measures of chaos and spectral decomposition of dynamical systems on the interval,, Trans. Amer. Math. Soc., 344 (1994), 737.  doi: 10.2307/2154504.  Google Scholar

[21]

J. C. Xiong, A chaotic map with topological entropy,, Acta Math. Sci. (English Ed.), 6 (1986), 439.   Google Scholar

[1]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[2]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[3]

Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083

[4]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275

[5]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[6]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[7]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[8]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

[9]

Dominik Kwietniak. Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2451-2467. doi: 10.3934/dcds.2013.33.2451

[10]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427

[11]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35

[12]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[13]

J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397

[14]

Evgeniy Timofeev, Alexei Kaltchenko. Nearest-neighbor entropy estimators with weak metrics. Advances in Mathematics of Communications, 2014, 8 (2) : 119-127. doi: 10.3934/amc.2014.8.119

[15]

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

[16]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[17]

John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293

[18]

Sergio Muñoz. Robust transitivity of maps of the real line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1163-1177. doi: 10.3934/dcds.2015.35.1163

[19]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

[20]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (5)

[Back to Top]