American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 221-226. [2] Ethan Akin and Sergii Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. [3] F. Balibrea, J. Smítal and M. vStefánková, The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583. doi: 10.1016/j.chaos.2004.06.011. [4] John Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885. [5] J. Banks, T. T. D. Nguyen, P. Oprocha and B. Trotta, Dynamics of spacing shifts,, Discrete Continuous Dynam. Systems - A, (). [6] Vitaly Bergelson, Ergodic Ramsey theory, Logic and combinatorics (Arcata, Calif., 1985), 63-87, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, (1987). doi: 10.1090/conm/065/891243. [7] A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578. doi: 10.1090/S0002-9939-02-06349-9. [8] Tomasz Downarowicz, Positive topological entropy implies chaos DC$2$, to appear in Proc. Amer. Math. Soc., arXiv:1110.5201v1, (2011). doi: 10.1017/CBO9780511976155. [9] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. [10] Harry Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," Princeton University Press, Princeton, N.J., (1981), xi+203 pp. [11] Harry Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211-234. doi: 10.1090/S0273-0979-1981-14932-6. [12] L. Wayne Goodwyn, Some counter-examples in topological entropy, Topology, 11 (1972), 377-385. [13] Wen Huang, Hanfeng Li and Xiangdong Ye, Family-independence for topological and measurable dynamics, Trans. Amer. Math Soc., 364 (2012), 5209-5245. doi: 10.1090/S0002-9947-2012-05493-6. [14] Víctor Jiménez López and L'ubomir Snoha, Stroboscopical property, equicontinuity and weak mixing, Iteration theory (ECIT '02), 235-244, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, (2004). [15] David Kerr and Hanfeng Li, Independence in topological and $C*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z. [16] Igor Kříž, Large independent sets in shift-invariant graphs: solution of Bergelson's problem, Graphs Combin., 3 (1987), 145-158. doi: 10.1007/BF01788538. [17] Dominik Kwietniak and Piotr Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661-1672. [18] Kenneth Lau and Alan Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (): 307. [19] Jian Li, Transitive points via Furstenberg family, Topology and its Applications, 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013. [20] Jian Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286. doi: 10.4064/fm216-3-4. [21] Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. xvi+495 pp. doi: 10.1017/CBO9780511626302. [22] Jan de Vries, "Elements of Topological Dynamics," Mathematics and Its Applications, 257, Kluwer Academic Publishers Group, Dordrecht, 1993. xvi+748 pp. [23] Randall McCutcheon, Three results in recurrence, Ergodic Theory and Its Connections With Harmonic Analysis (Alexandria, 1993), 349-358, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, (1995). doi: 10.1017/CBO9780511574818.015. [24] Piotr Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7. [25] Piotr Oprocha, Minimal systems and distributionally scrambled sets,, preprint, (). [26] William Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. [27] Rafał Pikuła, On some notions of chaos in dimension zero, Colloq. Math., 107 (2007), 167-177. doi: 10.4064/cm107-2-1. [28] Alfred Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493. [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), 737-754. doi: 10.2307/2154504. [30] Paul C. Shields, "The Ergodic Theory of Discrete Sample Paths," Graduate Studies in Mathematics, 13, American Mathematical Society, Providence, RI, 1996. xii+249 pp. [31] Karl Sigmund, On the distribution of periodic points for $\beta$-shifts, Monatsh. Math., 82 (1976), 247-252. [32] Klaus Thomsen, On the structure of beta shifts in "Algebraic and Topological Dynamics" 321-332, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005). doi: 10.1090/conm/385/07204. [33] Xiangdong Ye and Ruifeng Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601-1620. doi: 10.1088/0951-7715/21/7/012. [34] Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. ix+250 pp.

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References:
 [1] Dawoud Ahmadi Dastjerdi and Maliheh Dabbaghian Amiri, Characterization of entropy for spacing shifts, Acta Math. Univ. Comenianae, LXXXI (2012), 221-226. [2] Ethan Akin and Sergii Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313. [3] F. Balibrea, J. Smítal and M. vStefánková, The three versions of distributional chaos, Chaos Solitons Fractals, 23 (2005), 1581-1583. doi: 10.1016/j.chaos.2004.06.011. [4] John Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885. [5] J. Banks, T. T. D. Nguyen, P. Oprocha and B. Trotta, Dynamics of spacing shifts,, Discrete Continuous Dynam. Systems - A, (). [6] Vitaly Bergelson, Ergodic Ramsey theory, Logic and combinatorics (Arcata, Calif., 1985), 63-87, Contemp. Math., 65, Amer. Math. Soc., Providence, RI, (1987). doi: 10.1090/conm/065/891243. [7] A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578. doi: 10.1090/S0002-9939-02-06349-9. [8] Tomasz Downarowicz, Positive topological entropy implies chaos DC$2$, to appear in Proc. Amer. Math. Soc., arXiv:1110.5201v1, (2011). doi: 10.1017/CBO9780511976155. [9] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49. [10] Harry Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," Princeton University Press, Princeton, N.J., (1981), xi+203 pp. [11] Harry Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211-234. doi: 10.1090/S0273-0979-1981-14932-6. [12] L. Wayne Goodwyn, Some counter-examples in topological entropy, Topology, 11 (1972), 377-385. [13] Wen Huang, Hanfeng Li and Xiangdong Ye, Family-independence for topological and measurable dynamics, Trans. Amer. Math Soc., 364 (2012), 5209-5245. doi: 10.1090/S0002-9947-2012-05493-6. [14] Víctor Jiménez López and L'ubomir Snoha, Stroboscopical property, equicontinuity and weak mixing, Iteration theory (ECIT '02), 235-244, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, (2004). [15] David Kerr and Hanfeng Li, Independence in topological and $C*$-dynamics, Math. Ann., 338 (2007), 869-926. doi: 10.1007/s00208-007-0097-z. [16] Igor Kříž, Large independent sets in shift-invariant graphs: solution of Bergelson's problem, Graphs Combin., 3 (1987), 145-158. doi: 10.1007/BF01788538. [17] Dominik Kwietniak and Piotr Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661-1672. [18] Kenneth Lau and Alan Zame, On weak mixing of cascades,, Math. Systems Theory, 6 (): 307. [19] Jian Li, Transitive points via Furstenberg family, Topology and its Applications, 158 (2011), 2221-2231. doi: 10.1016/j.topol.2011.07.013. [20] Jian Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286. doi: 10.4064/fm216-3-4. [21] Douglas Lind and Brian Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. xvi+495 pp. doi: 10.1017/CBO9780511626302. [22] Jan de Vries, "Elements of Topological Dynamics," Mathematics and Its Applications, 257, Kluwer Academic Publishers Group, Dordrecht, 1993. xvi+748 pp. [23] Randall McCutcheon, Three results in recurrence, Ergodic Theory and Its Connections With Harmonic Analysis (Alexandria, 1993), 349-358, London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, (1995). doi: 10.1017/CBO9780511574818.015. [24] Piotr Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7. [25] Piotr Oprocha, Minimal systems and distributionally scrambled sets,, preprint, (). [26] William Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. [27] Rafał Pikuła, On some notions of chaos in dimension zero, Colloq. Math., 107 (2007), 167-177. doi: 10.4064/cm107-2-1. [28] Alfred Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493. [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), 737-754. doi: 10.2307/2154504. [30] Paul C. Shields, "The Ergodic Theory of Discrete Sample Paths," Graduate Studies in Mathematics, 13, American Mathematical Society, Providence, RI, 1996. xii+249 pp. [31] Karl Sigmund, On the distribution of periodic points for $\beta$-shifts, Monatsh. Math., 82 (1976), 247-252. [32] Klaus Thomsen, On the structure of beta shifts in "Algebraic and Topological Dynamics" 321-332, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, (2005). doi: 10.1090/conm/385/07204. [33] Xiangdong Ye and Ruifeng Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601-1620. doi: 10.1088/0951-7715/21/7/012. [34] Peter Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. ix+250 pp.
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