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 and Continuous Dynamical Systems, 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), 1421-1433. doi: 10.1088/0951-7715/16/4/313.

[2]

E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems, in "Chapel Hill Ergodic Theory Workshops," Contemp. Math., 356, Amer. Math. Soc., Providence, RI, (2004), 21-79. doi: 10.1090/conm/356/06496.

[3]

L. Alseda, M. del Rio and J. Rodriguez, Transitivity and dense periodicity for graph maps, J. Diff. Eq. Applications, 9 (2003), 577-598. doi: 10.1080/1023619021000040515.

[4]

F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361. doi: 10.4064/cm110-2-3.

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885.

[6]

J. Banks and B. Trotta, Weak mixing implies mixing for maps on topological graphs, J. Diff. Eq. Applications, 11 (2005), 1071-1080. doi: 10.1080/1023619050029557.

[7]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68. doi: 10.1515/crll.2002.053.

[8]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.

[10]

H. Furstenberg, Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.

[11]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[12]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272. doi: 10.1016/S0166-8641(01)00025-6.

[13]

A. Iwanik, Independence and scrambled sets for chaotic mappings, in "The Mathematical Heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, (1991), 372-378.

[14]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z.

[15]

K. Lau and A. Zame, On weak mixing of cascades, Math. Systems Theory, 6 (1972/73), 307-311.

[16]

G. Liao, Z. Chu and Q. Fan, Relations between mixing and distributional chaos, Chaos Solitons Fractals, 41 (2009), 1994-2000. doi: 10.1016/j.chaos.2008.08.003.

[17]

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

[18]

Edwin E. Moise, "Geometric Topology in Dimensions $2$ and $3$," Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977.

[19]

P. Oprocha, Relations between distributional and Devaney chaos, Chaos, 16 (2006), 033112, 5 pp. doi: 10.1063/1.2225513.

[20]

B. Schweizer and J. Smítal, Measures of chaos and spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504.

[21]

J. C. Xiong, A chaotic map with topological entropy, Acta Math. Sci. (English Ed.), 6 (1986), 439-443.

show all references

References:
[1]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313.

[2]

E. Akin, Lectures on Cantor and Mycielski sets for dynamical systems, in "Chapel Hill Ergodic Theory Workshops," Contemp. Math., 356, Amer. Math. Soc., Providence, RI, (2004), 21-79. doi: 10.1090/conm/356/06496.

[3]

L. Alseda, M. del Rio and J. Rodriguez, Transitivity and dense periodicity for graph maps, J. Diff. Eq. Applications, 9 (2003), 577-598. doi: 10.1080/1023619021000040515.

[4]

F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361. doi: 10.4064/cm110-2-3.

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885.

[6]

J. Banks and B. Trotta, Weak mixing implies mixing for maps on topological graphs, J. Diff. Eq. Applications, 11 (2005), 1071-1080. doi: 10.1080/1023619050029557.

[7]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68. doi: 10.1515/crll.2002.053.

[8]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.

[9]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.

[10]

H. Furstenberg, Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.

[11]

H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.

[12]

W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272. doi: 10.1016/S0166-8641(01)00025-6.

[13]

A. Iwanik, Independence and scrambled sets for chaotic mappings, in "The Mathematical Heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, (1991), 372-378.

[14]

D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z.

[15]

K. Lau and A. Zame, On weak mixing of cascades, Math. Systems Theory, 6 (1972/73), 307-311.

[16]

G. Liao, Z. Chu and Q. Fan, Relations between mixing and distributional chaos, Chaos Solitons Fractals, 41 (2009), 1994-2000. doi: 10.1016/j.chaos.2008.08.003.

[17]

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

[18]

Edwin E. Moise, "Geometric Topology in Dimensions $2$ and $3$," Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977.

[19]

P. Oprocha, Relations between distributional and Devaney chaos, Chaos, 16 (2006), 033112, 5 pp. doi: 10.1063/1.2225513.

[20]

B. Schweizer and J. Smítal, Measures of chaos and spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504.

[21]

J. C. Xiong, A chaotic map with topological entropy, Acta Math. Sci. (English Ed.), 6 (1986), 439-443.

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