June  2013, 33(6): 2451-2467. doi: 10.3934/dcds.2013.33.2451

Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts

1. 

Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received  February 2012 Revised  October 2012 Published  December 2012

Positive topological entropy and distributional chaos are characterized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC$2$-chaotic (or equivalently, DC$3$-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC$1$-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this characterization. Moreover, it is shown that a spacing shift $\Omega_P$ has positive topological entropy if and only if $\mathbb{N}\setminus P$ is a set of Poincaré recurrence. Using a result of Kříž an example of a proximal spacing shift with positive entropy is constructed. Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results on difference sets.
Citation: 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
References:
[1]

Dawoud Ahmadi Dastjerdi and Maliheh Dabbaghian Amiri, Characterization of entropy for spacing shifts,, Acta Math. Univ. Comenianae, LXXXI (2012), 221. Google Scholar

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

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

[5]

J. Banks, T. T. D. Nguyen, P. Oprocha and B. Trotta, Dynamics of spacing shifts,, Discrete Continuous Dynam. Systems - A, (). Google Scholar

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Vitaly Bergelson, Ergodic Ramsey theory,, Logic and combinatorics (Arcata, 65 (1987), 63. doi: 10.1090/conm/065/891243. Google Scholar

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A. Blokh and A. Fieldsteel, Sets that force recurrence,, Proc. Amer. Math. Soc., 130 (2002), 3571. doi: 10.1090/S0002-9939-02-06349-9. Google Scholar

[8]

Tomasz Downarowicz, Positive topological entropy implies chaos DC$2$,, to appear in Proc. Amer. Math. Soc., (2011). doi: 10.1017/CBO9780511976155. Google Scholar

[9]

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

[10]

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

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Harry Furstenberg, Poincaré recurrence and number theory,, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211. doi: 10.1090/S0273-0979-1981-14932-6. Google Scholar

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L. Wayne Goodwyn, Some counter-examples in topological entropy,, Topology, 11 (1972), 377. Google Scholar

[13]

Wen Huang, Hanfeng Li and Xiangdong Ye, Family-independence for topological and measurable dynamics,, Trans. Amer. Math Soc., 364 (2012), 5209. doi: 10.1090/S0002-9947-2012-05493-6. Google Scholar

[14]

Víctor Jiménez López and L'ubomir Snoha, Stroboscopical property, equicontinuity and weak mixing,, Iteration theory (ECIT '02), 346 (2004), 235. Google Scholar

[15]

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

[16]

Igor Kříž, Large independent sets in shift-invariant graphs: solution of Bergelson's problem,, Graphs Combin., 3 (1987), 145. doi: 10.1007/BF01788538. Google Scholar

[17]

Dominik Kwietniak and Piotr Oprocha, On weak mixing, minimality and weak disjointness of all iterates,, Erg. Th. Dynam. Syst., 32 (2012), 1661. Google Scholar

[18]

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

[19]

Jian Li, Transitive points via Furstenberg family,, Topology and its Applications, 158 (2011), 2221. doi: 10.1016/j.topol.2011.07.013. Google Scholar

[20]

Jian Li, Dynamical characterization of C-sets and its application,, Fund. Math., 216 (2012), 259. doi: 10.4064/fm216-3-4. Google Scholar

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

[22]

Jan de Vries, "Elements of Topological Dynamics,", Mathematics and Its Applications, 257 (1993). Google Scholar

[23]

Randall McCutcheon, Three results in recurrence,, Ergodic Theory and Its Connections With Harmonic Analysis (Alexandria, 205 (1995), 349. doi: 10.1017/CBO9780511574818.015. Google Scholar

[24]

Piotr Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901. doi: 10.1090/S0002-9947-09-04810-7. Google Scholar

[25]

Piotr Oprocha, Minimal systems and distributionally scrambled sets,, preprint, (). Google Scholar

[26]

William Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. Google Scholar

[27]

Rafał Pikuła, On some notions of chaos in dimension zero,, Colloq. Math., 107 (2007), 167. doi: 10.4064/cm107-2-1. Google Scholar

[28]

Alfred Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar., 8 (1957), 477. Google Scholar

[29]

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

[30]

Paul C. Shields, "The Ergodic Theory of Discrete Sample Paths,", Graduate Studies in Mathematics, 13 (1996). Google Scholar

[31]

Karl Sigmund, On the distribution of periodic points for $\beta $-shifts,, Monatsh. Math., 82 (1976), 247. Google Scholar

[32]

Klaus Thomsen, On the structure of beta shifts, in, 385 (2005), 321. doi: 10.1090/conm/385/07204. Google Scholar

[33]

Xiangdong Ye and Ruifeng Zhang, On sensitive sets in topological dynamics,, Nonlinearity, 21 (2008), 1601. doi: 10.1088/0951-7715/21/7/012. Google Scholar

[34]

Peter Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar

show all references

References:
[1]

Dawoud Ahmadi Dastjerdi and Maliheh Dabbaghian Amiri, Characterization of entropy for spacing shifts,, Acta Math. Univ. Comenianae, LXXXI (2012), 221. Google Scholar

[2]

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

[3]

F. Balibrea, J. Smítal and M. vStefánková, The three versions of distributional chaos,, Chaos Solitons Fractals, 23 (2005), 1581. doi: 10.1016/j.chaos.2004.06.011. Google Scholar

[4]

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

[5]

J. Banks, T. T. D. Nguyen, P. Oprocha and B. Trotta, Dynamics of spacing shifts,, Discrete Continuous Dynam. Systems - A, (). Google Scholar

[6]

Vitaly Bergelson, Ergodic Ramsey theory,, Logic and combinatorics (Arcata, 65 (1987), 63. doi: 10.1090/conm/065/891243. Google Scholar

[7]

A. Blokh and A. Fieldsteel, Sets that force recurrence,, Proc. Amer. Math. Soc., 130 (2002), 3571. doi: 10.1090/S0002-9939-02-06349-9. Google Scholar

[8]

Tomasz Downarowicz, Positive topological entropy implies chaos DC$2$,, to appear in Proc. Amer. Math. Soc., (2011). doi: 10.1017/CBO9780511976155. Google Scholar

[9]

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

[10]

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

[11]

Harry Furstenberg, Poincaré recurrence and number theory,, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211. doi: 10.1090/S0273-0979-1981-14932-6. Google Scholar

[12]

L. Wayne Goodwyn, Some counter-examples in topological entropy,, Topology, 11 (1972), 377. Google Scholar

[13]

Wen Huang, Hanfeng Li and Xiangdong Ye, Family-independence for topological and measurable dynamics,, Trans. Amer. Math Soc., 364 (2012), 5209. doi: 10.1090/S0002-9947-2012-05493-6. Google Scholar

[14]

Víctor Jiménez López and L'ubomir Snoha, Stroboscopical property, equicontinuity and weak mixing,, Iteration theory (ECIT '02), 346 (2004), 235. Google Scholar

[15]

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

[16]

Igor Kříž, Large independent sets in shift-invariant graphs: solution of Bergelson's problem,, Graphs Combin., 3 (1987), 145. doi: 10.1007/BF01788538. Google Scholar

[17]

Dominik Kwietniak and Piotr Oprocha, On weak mixing, minimality and weak disjointness of all iterates,, Erg. Th. Dynam. Syst., 32 (2012), 1661. Google Scholar

[18]

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

[19]

Jian Li, Transitive points via Furstenberg family,, Topology and its Applications, 158 (2011), 2221. doi: 10.1016/j.topol.2011.07.013. Google Scholar

[20]

Jian Li, Dynamical characterization of C-sets and its application,, Fund. Math., 216 (2012), 259. doi: 10.4064/fm216-3-4. Google Scholar

[21]

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

[22]

Jan de Vries, "Elements of Topological Dynamics,", Mathematics and Its Applications, 257 (1993). Google Scholar

[23]

Randall McCutcheon, Three results in recurrence,, Ergodic Theory and Its Connections With Harmonic Analysis (Alexandria, 205 (1995), 349. doi: 10.1017/CBO9780511574818.015. Google Scholar

[24]

Piotr Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901. doi: 10.1090/S0002-9947-09-04810-7. Google Scholar

[25]

Piotr Oprocha, Minimal systems and distributionally scrambled sets,, preprint, (). Google Scholar

[26]

William Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. Google Scholar

[27]

Rafał Pikuła, On some notions of chaos in dimension zero,, Colloq. Math., 107 (2007), 167. doi: 10.4064/cm107-2-1. Google Scholar

[28]

Alfred Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar., 8 (1957), 477. Google Scholar

[29]

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

[30]

Paul C. Shields, "The Ergodic Theory of Discrete Sample Paths,", Graduate Studies in Mathematics, 13 (1996). Google Scholar

[31]

Karl Sigmund, On the distribution of periodic points for $\beta $-shifts,, Monatsh. Math., 82 (1976), 247. Google Scholar

[32]

Klaus Thomsen, On the structure of beta shifts, in, 385 (2005), 321. doi: 10.1090/conm/385/07204. Google Scholar

[33]

Xiangdong Ye and Ruifeng Zhang, On sensitive sets in topological dynamics,, Nonlinearity, 21 (2008), 1601. doi: 10.1088/0951-7715/21/7/012. Google Scholar

[34]

Peter Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982). Google Scholar

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