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

Li-Yorke Chaos for ultragraph shift spaces

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.

Citation: Daniel Gonçalves, Bruno Brogni Uggioni. Li-Yorke Chaos for ultragraph shift spaces. Discrete and Continuous Dynamical Systems, 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.

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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. 

[26]

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

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[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. 

[26]

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

[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.

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