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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[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).

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

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.

[2]

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

[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.

[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.

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[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).

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

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