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April  2020, 40(4): 2347-2365. doi: 10.3934/dcds.2020117

Li-Yorke Chaos for ultragraph shift spaces

Daniel Gonçalves 1,, and  Bruno Brogni Uggioni 2,

1. 

UFSC – Departamento de Matemática, Florianópolis, SC 88040-900, Brazil
2. 

IFRS – Campus Canoas, Canoas, RS 92412-240, Brazil

* Corresponding author: Daniel Gonçalves

Received  July 2019 Published  January 2020

Recently, in connection with C*-algebra theory, the first author and Danilo Royer introduced ultragraph shift spaces. In this paper we define a family of metrics for the topology in such spaces, and use these metrics to study the existence of chaos in the shift. In particular we characterize all ultragraph shift spaces that have Li-Yorke chaos (an uncountable scrambled set), and prove that such property implies the existence of a perfect and scrambled set in the ultragraph shift space. Furthermore, this scrambled set can be chosen compact, which is not the case for a labelled edge shift (with the product topology) of an infinite graph.

Keywords: Symbolic dynamics, Li-Yorke Chaos, ultragraph edge shift space, infinite alphabet.
Mathematics Subject Classification: Primary: 37B10, 37B20; Secondary: 37D40, 54H20.
Citation: Daniel Gonçalves, Bruno Brogni Uggioni. Li-Yorke Chaos for ultragraph shift spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2347-2365. doi: 10.3934/dcds.2020117
References:
[1]

G. Abrams, P. Ara and M. S. Molina, Leavitt Path Algebras, Lecture Notes in Mathematics, 2191. Springer, London, 2017.  Google Scholar

[2]

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

[3]

G. G. de Castro and D. Gonçalves, KMS and ground states on ultragraph C*-algebras, Integral Equations Operator Theory, 90, (2018), Art. 63, 23 pp. doi: 10.1007/s00020-018-2490-2.  Google Scholar

[4]

T. Ceccherini-Silberstein and M. Coornaert, A generalization of the Curtis-Hedlund theorem, Theoret. Comput. Sci, 400, (2008), 225–229. doi: 10.1016/j.tcs.2008.02.050.  Google Scholar

[5]

X. P. Dai and X. J. Tang, Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics, J Differential Equations, 263 (2017), 5521-5553.  doi: 10.1016/j.jde.2017.06.021.  Google Scholar

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T. Downarowicz and Y. Lacroix, Measure-theoretic chaos, Ergodic Theory Dynam. Systems, 34 (2014), 110-131.  doi: 10.1017/etds.2012.117.  Google Scholar

[7]

G. Edgar, Measure, Topology, and Fractal Geometry, Second edition, Undergraduate Texts in Mathematics, Springer, New York, 2008. doi: 10.1007/978-0-387-74749-1.  Google Scholar

[8]

F. Garcia-Ramos and L. Jin, Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc., 145 (2017), 2959-2969.  doi: 10.1090/proc/13440.  Google Scholar

[9]

D. Gonçalves and D. Royer, Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras, Int. Math. Res. Not. IMRN, 2019 (2019), 2177-2203.  doi: 10.1093/imrn/rnx175.  Google Scholar

[10]

D. Gonçalves and D. Royer, $(M+1)$-step shift spaces that are not conjugate to $M$-step shift spaces, Bull. Sci. Math., 139 (2015), 178-183.  doi: 10.1016/j.bulsci.2014.08.007.  Google Scholar

[11]

D. Gonçalves and D. Royer, Ultragraphs and shift spaces over infinite alphabets, Bull. Sci. Math., 141 (2017), 25-45.  doi: 10.1016/j.bulsci.2016.10.002.  Google Scholar

[12]

D. Gonçalves and M. Sobottka, Continuous shift commuting maps between ultragraph shift spaces, Discrete Contin. Dyn. Syst., 39 (2019), 1033-1048.  doi: 10.3934/dcds.2019043.  Google Scholar

[13]

D. Gonçalves, M. Sobottka and C. Starling, Inverse semigroup shifts over countable alphabets, Semigroup Forum, 96 (2018), 203-240.  doi: 10.1007/s00233-017-9858-5.  Google Scholar

[14]

D. Gonçalves, M. Sobottka and C. Starling, Sliding block codes between shift spaces over infinite alphabets, Math. Nachr., 289 (2016), 2178-2191.  doi: 10.1002/mana.201500309.  Google Scholar

[15]

D. Gonçalves, M. Sobottka and C. Starling, Two-sided shift spaces over infinite alphabets, J. Aust. Math. Soc., 103 (2017), 357-386.  doi: 10.1017/S1446788717000039.  Google Scholar

[16]

D. Gonçalves and B. B. Uggioni, Ultragraph shift spaces and chaos, Bull. Sci. math., 158 (2019), 102807, 23 pp. doi: 10.1016/j.bulsci.2019.102807.  Google Scholar

[17]

S. F. Kolyada, Li-Yorke sensitivity and other concepts of chaos, Ukrainian Math. J., 56 (2004), 1242-1257.  doi: 10.1007/s11253-005-0055-4.  Google Scholar

[18]

T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[19]

A. E. Marrero and P. S. Muhly, Groupoid and inverse semigroup presentations of ultragraph $C^*$-algebras, Semigroup Forum, 77 (2008), 399-422.  doi: 10.1007/s00233-008-9046-8.  Google Scholar

[20]

P. Oprocha and P. Wilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals, 31 (2007), 347-355.  doi: 10.1016/j.chaos.2005.09.069.  Google Scholar

[21]

W. Ott, M. Tomforde and P. N. Willis, One-sided shift spaces over infinite alphabets, New York J. Math., NYJM Monographs, State University of New York, University at Albany, Albany, NY, 5 (2014), 54 pp.  Google Scholar

[22]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448.  doi: 10.1017/S014338570000359X.  Google Scholar

[23]

B. E. Raines and T. Underwood, Scrambled sets in shift spaces on a countable alphabet, Proc. Amer. Math. Soc., 144 (2016), 217-224.  doi: 10.1090/proc/12690.  Google Scholar

[24]

I. A. Salama, Topological entropy and recurrence of countable chains, Pacific Journal of Mathematics, 134 (1988), 325-341.  doi: 10.2140/pjm.1988.134.325.  Google Scholar

[25]

M. Sobottka and D. Gonçalves, A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon theorem, J. Cell. Autom., 12 (2017), 209-215.   Google Scholar

[26]

M. Tomforde, A unified approch to Exel-Laca algebras and $C^*$-algebras associated to graphs, J. Operator Theory, 50 (2003), 345-368.   Google Scholar

[27]

S. B. G. Webster, The path space of a directed graph, Proc. Amer. Math. Soc., 142 (2014), 213-225.  doi: 10.1090/S0002-9939-2013-11755-7.  Google Scholar

show all references

References:
[1]

G. Abrams, P. Ara and M. S. Molina, Leavitt Path Algebras, Lecture Notes in Mathematics, 2191. Springer, London, 2017.  Google Scholar

[2]

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

[3]

G. G. de Castro and D. Gonçalves, KMS and ground states on ultragraph C*-algebras, Integral Equations Operator Theory, 90, (2018), Art. 63, 23 pp. doi: 10.1007/s00020-018-2490-2.  Google Scholar

[4]

T. Ceccherini-Silberstein and M. Coornaert, A generalization of the Curtis-Hedlund theorem, Theoret. Comput. Sci, 400, (2008), 225–229. doi: 10.1016/j.tcs.2008.02.050.  Google Scholar

[5]

X. P. Dai and X. J. Tang, Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos for topological dynamics, J Differential Equations, 263 (2017), 5521-5553.  doi: 10.1016/j.jde.2017.06.021.  Google Scholar

[6]

T. Downarowicz and Y. Lacroix, Measure-theoretic chaos, Ergodic Theory Dynam. Systems, 34 (2014), 110-131.  doi: 10.1017/etds.2012.117.  Google Scholar

[7]

G. Edgar, Measure, Topology, and Fractal Geometry, Second edition, Undergraduate Texts in Mathematics, Springer, New York, 2008. doi: 10.1007/978-0-387-74749-1.  Google Scholar

[8]

F. Garcia-Ramos and L. Jin, Mean proximality and mean Li-Yorke chaos, Proc. Amer. Math. Soc., 145 (2017), 2959-2969.  doi: 10.1090/proc/13440.  Google Scholar

[9]

D. Gonçalves and D. Royer, Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras, Int. Math. Res. Not. IMRN, 2019 (2019), 2177-2203.  doi: 10.1093/imrn/rnx175.  Google Scholar

[10]

D. Gonçalves and D. Royer, $(M+1)$-step shift spaces that are not conjugate to $M$-step shift spaces, Bull. Sci. Math., 139 (2015), 178-183.  doi: 10.1016/j.bulsci.2014.08.007.  Google Scholar

[11]

D. Gonçalves and D. Royer, Ultragraphs and shift spaces over infinite alphabets, Bull. Sci. Math., 141 (2017), 25-45.  doi: 10.1016/j.bulsci.2016.10.002.  Google Scholar

[12]

D. Gonçalves and M. Sobottka, Continuous shift commuting maps between ultragraph shift spaces, Discrete Contin. Dyn. Syst., 39 (2019), 1033-1048.  doi: 10.3934/dcds.2019043.  Google Scholar

[13]

D. Gonçalves, M. Sobottka and C. Starling, Inverse semigroup shifts over countable alphabets, Semigroup Forum, 96 (2018), 203-240.  doi: 10.1007/s00233-017-9858-5.  Google Scholar

[14]

D. Gonçalves, M. Sobottka and C. Starling, Sliding block codes between shift spaces over infinite alphabets, Math. Nachr., 289 (2016), 2178-2191.  doi: 10.1002/mana.201500309.  Google Scholar

[15]

D. Gonçalves, M. Sobottka and C. Starling, Two-sided shift spaces over infinite alphabets, J. Aust. Math. Soc., 103 (2017), 357-386.  doi: 10.1017/S1446788717000039.  Google Scholar

[16]

D. Gonçalves and B. B. Uggioni, Ultragraph shift spaces and chaos, Bull. Sci. math., 158 (2019), 102807, 23 pp. doi: 10.1016/j.bulsci.2019.102807.  Google Scholar

[17]

S. F. Kolyada, Li-Yorke sensitivity and other concepts of chaos, Ukrainian Math. J., 56 (2004), 1242-1257.  doi: 10.1007/s11253-005-0055-4.  Google Scholar

[18]

T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[19]

A. E. Marrero and P. S. Muhly, Groupoid and inverse semigroup presentations of ultragraph $C^*$-algebras, Semigroup Forum, 77 (2008), 399-422.  doi: 10.1007/s00233-008-9046-8.  Google Scholar

[20]

P. Oprocha and P. Wilczyński, Shift spaces and distributional chaos, Chaos Solitons Fractals, 31 (2007), 347-355.  doi: 10.1016/j.chaos.2005.09.069.  Google Scholar

[21]

W. Ott, M. Tomforde and P. N. Willis, One-sided shift spaces over infinite alphabets, New York J. Math., NYJM Monographs, State University of New York, University at Albany, Albany, NY, 5 (2014), 54 pp.  Google Scholar

[22]

K. Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems, 6 (1986), 415-448.  doi: 10.1017/S014338570000359X.  Google Scholar

[23]

B. E. Raines and T. Underwood, Scrambled sets in shift spaces on a countable alphabet, Proc. Amer. Math. Soc., 144 (2016), 217-224.  doi: 10.1090/proc/12690.  Google Scholar

[24]

I. A. Salama, Topological entropy and recurrence of countable chains, Pacific Journal of Mathematics, 134 (1988), 325-341.  doi: 10.2140/pjm.1988.134.325.  Google Scholar

[25]

M. Sobottka and D. Gonçalves, A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon theorem, J. Cell. Autom., 12 (2017), 209-215.   Google Scholar

[26]

M. Tomforde, A unified approch to Exel-Laca algebras and $C^*$-algebras associated to graphs, J. Operator Theory, 50 (2003), 345-368.   Google Scholar

[27]

S. B. G. Webster, The path space of a directed graph, Proc. Amer. Math. Soc., 142 (2014), 213-225.  doi: 10.1090/S0002-9939-2013-11755-7.  Google Scholar

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