December  2015, 20(10): 3487-3505. doi: 10.3934/dcdsb.2015.20.3487

Two results on entropy, chaos and independence in symbolic dynamics

1. 

Department of Mathematics, Cracow University of Economics, ul. Rakowicka 27, 31-510 Kraków, Poland

2. 

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

3. 

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

4. 

Department of Mathematics, Shantou University, Shantou, Guangdong 515063

Received  December 2014 Revised  March 2015 Published  September 2015

We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics:
    1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces.
    2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair.
Our proofs are new and yield conclusions stronger than what was known before.
Citation: Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[2]

R. P. Anstee, L. Rónyai and A. Sali, Shattering news,, Graphs Combin., 18 (2002), 59. doi: 10.1007/s003730200003. Google Scholar

[3]

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

[4]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332. doi: 10.2307/2324899. Google Scholar

[5]

F. Blanchard, Topological chaos: What may this mean?,, J. Difference Equ. Appl., 15 (2009), 23. doi: 10.1080/10236190802385355. Google Scholar

[6]

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

[7]

F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275. Google Scholar

[8]

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

[9]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (2003). Google Scholar

[10]

T. Downarowicz, Positive topological entropy implies chaos DC2,, Proc. Amer. Math. Soc., 142 (2014), 137. doi: 10.1090/S0002-9939-2013-11717-X. Google Scholar

[11]

T. Downarowicz and X. Ye, When every point is either transitive or periodic,, Colloq. Math., 93 (2002), 137. doi: 10.4064/cm93-1-9. Google Scholar

[12]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar

[13]

E. Glasner and X. Ye, Local entropy theory,, Ergodic Theory Dynam. Systems, 29 (2009), 321. doi: 10.1017/S0143385708080309. Google Scholar

[14]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067. doi: 10.1088/0951-7715/6/6/014. Google Scholar

[15]

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems,, J. Amer. Math. Soc., 8 (1995), 665. doi: 10.2307/2152926. Google Scholar

[16]

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

[17]

W. Huang and X. Ye, A local variational relation and applications,, Israel J. Math., 151 (2006), 237. doi: 10.1007/BF02777364. Google Scholar

[18]

W. Huang, J. Li and X. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems,, J. Funct. Anal., 266 (2014), 3377. doi: 10.1016/j.jfa.2014.01.005. Google Scholar

[19]

M. G. Karpovsky and V. D. Milman, Coordinate density of sets of vectors,, Discrete Math., 24 (1978), 177. doi: 10.1016/0012-365X(78)90197-8. Google Scholar

[20]

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

[21]

P. Komjáth and V. Totik, Problems and Theorems in Classical Set Theory,, Problem Books in Mathematics, (2006). Google Scholar

[22]

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

[23]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics,, to appear in Acta Math. Sin. (Engl. Ser.), (2015). doi: 10.1007/s10114-015-4574-0. Google Scholar

[24]

S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243. doi: 10.2307/2154217. Google Scholar

[25]

S. H. Li, Dynamical properties of the shift maps on the inverse limit spaces,, Ergodic Theory Dynam. Systems, 12 (1992), 95. doi: 10.1017/S0143385700006611. Google Scholar

[26]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254. Google Scholar

[27]

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

[28]

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

[29]

A. Pajor, Sous-espaces $l_1^n$ des Espaces de Banach,, (French) [$l_1^n$-subspaces of Banach spaces] with an introduction by Gilles Pisier, (1985). Google Scholar

[30]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow,, Preprint, (2014). Google Scholar

[31]

Y. Peres, A combinatorial application of the maximal ergodic theorem,, Bull. London Math. Soc., 20 (1988), 248. doi: 10.1112/blms/20.3.248. Google Scholar

[32]

R. Pikuła, On enveloping semigroups of almost one-to-one extensions of minimal group rotations,, Colloq. Math., 129 (2012), 249. doi: 10.4064/cm129-2-6. Google Scholar

[33]

I. Z. Ruzsa, On difference sets,, Studia Sci. Math. Hungar., 13 (1978), 319. Google Scholar

[34]

N. Sauer, On the density of families of sets,, J. Combinatorial Theory Ser. A, 13 (1972), 145. doi: 10.1016/0097-3165(72)90019-2. Google Scholar

[35]

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: 10.1090/S0002-9947-1994-1227094-X. Google Scholar

[36]

S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages,, Pacific J. Math., 41 (1972), 247. doi: 10.2140/pjm.1972.41.247. Google Scholar

[37]

J. Smítal, Chaotic functions with zero topological entropy,, Trans. Amer. Math. Soc., 297 (1986), 269. doi: 10.1090/S0002-9947-1986-0849479-9. Google Scholar

[38]

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

[39]

B. Weiss, Topological transitivity and ergodic measures,, Theory of Computing Systems, 5 (1971), 71. doi: 10.1007/BF01691469. Google Scholar

[40]

B. Weiss, Single Orbit Dynamics,, CBMS Regional Conference Series in Mathematics, 95 (2000). Google Scholar

[41]

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

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar

[2]

R. P. Anstee, L. Rónyai and A. Sali, Shattering news,, Graphs Combin., 18 (2002), 59. doi: 10.1007/s003730200003. Google Scholar

[3]

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

[4]

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332. doi: 10.2307/2324899. Google Scholar

[5]

F. Blanchard, Topological chaos: What may this mean?,, J. Difference Equ. Appl., 15 (2009), 23. doi: 10.1080/10236190802385355. Google Scholar

[6]

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

[7]

F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275. Google Scholar

[8]

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

[9]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (2003). Google Scholar

[10]

T. Downarowicz, Positive topological entropy implies chaos DC2,, Proc. Amer. Math. Soc., 142 (2014), 137. doi: 10.1090/S0002-9939-2013-11717-X. Google Scholar

[11]

T. Downarowicz and X. Ye, When every point is either transitive or periodic,, Colloq. Math., 93 (2002), 137. doi: 10.4064/cm93-1-9. Google Scholar

[12]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar

[13]

E. Glasner and X. Ye, Local entropy theory,, Ergodic Theory Dynam. Systems, 29 (2009), 321. doi: 10.1017/S0143385708080309. Google Scholar

[14]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067. doi: 10.1088/0951-7715/6/6/014. Google Scholar

[15]

E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems,, J. Amer. Math. Soc., 8 (1995), 665. doi: 10.2307/2152926. Google Scholar

[16]

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

[17]

W. Huang and X. Ye, A local variational relation and applications,, Israel J. Math., 151 (2006), 237. doi: 10.1007/BF02777364. Google Scholar

[18]

W. Huang, J. Li and X. Ye, Stable sets and mean Li-Yorke chaos in positive entropy systems,, J. Funct. Anal., 266 (2014), 3377. doi: 10.1016/j.jfa.2014.01.005. Google Scholar

[19]

M. G. Karpovsky and V. D. Milman, Coordinate density of sets of vectors,, Discrete Math., 24 (1978), 177. doi: 10.1016/0012-365X(78)90197-8. Google Scholar

[20]

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

[21]

P. Komjáth and V. Totik, Problems and Theorems in Classical Set Theory,, Problem Books in Mathematics, (2006). Google Scholar

[22]

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

[23]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics,, to appear in Acta Math. Sin. (Engl. Ser.), (2015). doi: 10.1007/s10114-015-4574-0. Google Scholar

[24]

S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243. doi: 10.2307/2154217. Google Scholar

[25]

S. H. Li, Dynamical properties of the shift maps on the inverse limit spaces,, Ergodic Theory Dynam. Systems, 12 (1992), 95. doi: 10.1017/S0143385700006611. Google Scholar

[26]

T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254. Google Scholar

[27]

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

[28]

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

[29]

A. Pajor, Sous-espaces $l_1^n$ des Espaces de Banach,, (French) [$l_1^n$-subspaces of Banach spaces] with an introduction by Gilles Pisier, (1985). Google Scholar

[30]

R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow,, Preprint, (2014). Google Scholar

[31]

Y. Peres, A combinatorial application of the maximal ergodic theorem,, Bull. London Math. Soc., 20 (1988), 248. doi: 10.1112/blms/20.3.248. Google Scholar

[32]

R. Pikuła, On enveloping semigroups of almost one-to-one extensions of minimal group rotations,, Colloq. Math., 129 (2012), 249. doi: 10.4064/cm129-2-6. Google Scholar

[33]

I. Z. Ruzsa, On difference sets,, Studia Sci. Math. Hungar., 13 (1978), 319. Google Scholar

[34]

N. Sauer, On the density of families of sets,, J. Combinatorial Theory Ser. A, 13 (1972), 145. doi: 10.1016/0097-3165(72)90019-2. Google Scholar

[35]

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: 10.1090/S0002-9947-1994-1227094-X. Google Scholar

[36]

S. Shelah, A combinatorial problem; stability and order for models and theories in infinitary languages,, Pacific J. Math., 41 (1972), 247. doi: 10.2140/pjm.1972.41.247. Google Scholar

[37]

J. Smítal, Chaotic functions with zero topological entropy,, Trans. Amer. Math. Soc., 297 (1986), 269. doi: 10.1090/S0002-9947-1986-0849479-9. Google Scholar

[38]

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

[39]

B. Weiss, Topological transitivity and ergodic measures,, Theory of Computing Systems, 5 (1971), 71. doi: 10.1007/BF01691469. Google Scholar

[40]

B. Weiss, Single Orbit Dynamics,, CBMS Regional Conference Series in Mathematics, 95 (2000). Google Scholar

[41]

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

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