Acta Mathematica Sinica, English Series

, Volume 32, Issue 1, pp 83–114 | Cite as

Recent development of chaos theory in topological dynamics

  • Jian LiEmail author
  • Xiang Dong Ye


We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li–Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.


Li–Yorke chaos Devaney chaos sensitive dependence on initial conditions distributional chaos weak mixing topological entropy Furstenberg family 

MR(2010) Subject Classification

54H20 37B05 37B40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adler, R. L., Konheim, A. G., McAndrew, M. H.: Topological entropy. Trans. Amer. Math. Soc., 114, 309–319 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Akin, E.: Recurrence in topological dynamics: Furstenberg families and Ellis actions. The University Series in Mathematics, Plenum Press, New York, 1997zbMATHCrossRefGoogle Scholar
  3. [3]
    Akin, E., Auslander, J., Berg, K.: When is a transitive map chaotic? In: Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., Vol. 5, de Gruyter, Berlin, 1996, 25–40Google Scholar
  4. [4]
    Akin, E., Glasner, E.: Residual properties and almost equicontinuity. J. Anal. Math., 84, 243–286 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Akin, E., Glasner, E., Huang, W., et al.: Sufficient conditions under which a transitive system is chaotic. Ergodic Theory Dynam. Systems, 30(5), 1277–1310 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Akin, E., Kolyada, S.: Li–Yorke sensitivity. Nonlinearity, 16(4), 1421–1433 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Auslander, J.: Minimal flows and their extensions. In: North-Holland Mathematics Studies, Vol. 153 North-Holland Publishing Co., Amsterdam, 1988Google Scholar
  8. [8]
    Auslander, J., Yorke, J. A.: Interval maps, factors of maps, and chaos. Tohoku Math. J. (2), 32(2), 177–188 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Balibrea, F., Guirao, J. L. G., Oprocha, P.: On invariant-scrambled sets. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20(9), 2925–2935 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Balibrea, F., Smítal, J., Stefánková, M.: The three versions of distributional chaos. Chaos Solitons Fractals, 23(5), 1581–1583 (2005)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals, 25(3), 681–685 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Banks, J., Brooks, J., Cairns, G., et al.: On Devaney’s definition of chaos. Amer. Math. Monthly, 99(4), 332–334 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math., 79, 81–92 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Blanchard, F.: Topological chaos: what may this mean? J. Difference Equ. Appl., 15(1), 23–46 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Blanchard, F., Glasner, E., Kolyada, S., et al.: On Li–Yorke pairs. J. Reine Angew. Math., 547, 51–68 (2002)MathSciNetzbMATHGoogle Scholar
  16. [16]
    Blanchard, F., Host, B., Ruette, S.: Asymptotic pairs in positive-entropy systems. Ergodic Theory Dynam. Systems, 22(3), 671–686 (2002)MathSciNetzbMATHGoogle Scholar
  17. [17]
    Blanchard, F., Huang, W.: Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin. Dyn. Syst., 20(2), 275–311 (2008)MathSciNetzbMATHGoogle Scholar
  18. [18]
    Blanchard, F., Huang, W., Snoha, L.: Topological size of scrambled sets. Colloq. Math., 110(2), 293–361 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Devaney, R. L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, second edition, 1989zbMATHGoogle Scholar
  20. [20]
    Dolezelová, J.: Scrambled and distributionally scrambled n-tuples. J. Difference Equ. Appl., 20(8), 1169–1177 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Downarowicz, T.: Positive topological entropy implies chaos DC2. Proc. Amer. Math. Soc., 142(1), 137–149 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Du, B.-S.: On the invariance of Li–Yorke chaos of interval maps. J. Difference Equ. Appl., 11(9), 823–828 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Fomin, S.: On dynamical systems with a purely point spectrum. Doklady Akad. Nauk SSSR (N.S.), 77, 29–32 (1951)zbMATHMathSciNetGoogle Scholar
  24. [24]
    Forys, M., Huang, W., Li, J., et al.: Invariant scrambled sets, uniform rigidity and weak mixing. Israel J. Math., to appear (2014)Google Scholar
  25. [25]
    Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981zbMATHCrossRefGoogle Scholar
  27. [27]
    García-Ramos, F.: Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Preprint, arXiv:1402.7327[math.DS]Google Scholar
  28. [28]
    Glasner, E.: Ergodic theory via joinings. In: Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003Google Scholar
  29. [29]
    Glasner, E., Weiss, B.: Sensitive dependence on initial conditions. Nonlinearity, 6(6), 1067–1075 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Glasner, E., Weiss, B.: Quasi-factors of zero-entropy systems. J. Amer. Math. Soc., 8(3), 665–686 (1995)MathSciNetzbMATHGoogle Scholar
  31. [31]
    Glasner, E., Ye, X.: Local entropy theory. Ergodic Theory Dynam. Systems, 29(2), 321–356 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Glasner, S.: Proximal flows. In: Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976Google Scholar
  33. [33]
    Glasner, S., Maon, D.: Rigidity in topological dynamics. Ergodic Theory Dynam. Systems, 9(2), 309–320 (1989)MathSciNetzbMATHGoogle Scholar
  34. [34]
    Gottschalk, W. H., Hedlund, G. A.: Topological dynamics. In: American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R. I., 1955Google Scholar
  35. [35]
    Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Comm. Math. Phys., 70(2), 133–160 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Guirao, J. L. G., Kwietniak, D., Lampart, M., et al.: Chaos on hyperspaces. Nonlinear Anal., 71(1–2), 1–8 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Huang, W.: Stable sets and -stable sets in positive-entropy systems. Comm. Math. Phys., 279(2), 535–557 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Huang, W., Kolyada, S., Zhang, G.: Multi-sensitivity, Lyapunov numbers and almost automorphic maps. Preprint (2014)Google Scholar
  39. [39]
    Huang, W., Li, H., Ye, X.: Family independence for topological and measurable dynamics. Trans. Amer. Math. Soc., 364(10), 5209–5242 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Huang, W., Li, J., Ye, X.: Stable sets and mean Li–Yorke chaos in positive entropy systems. J. Funct. Anal., 266(6), 3377–3394 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Huang, W., Li, J., Ye, X., Zhou, X.: Topological entropy and diagonal-weakly mixing sets. Preprint (2014)Google Scholar
  42. [42]
    Huang, W., Lu, P., Ye, X.: Measure-theoretical sensitivity and equicontinuity. Israel J. Math., 183, 233–283 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    Huang, W., Shao, S., Ye, X.: Mixing and proximal cells along sequences. Nonlinearity, 17(4), 1245–1260 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    Huang, W., Xu, L., Yi, Y.: Asymptotic pairs, stable sets and chaos in positive entropy systems. J. Funct. Anal., to appearGoogle Scholar
  45. [45]
    Huang, W., Ye, X.: Homeomorphisms with the whole compacta being scrambled sets. Ergodic Theory Dynam. Systems, 21(1), 77–91 (2001)MathSciNetzbMATHGoogle Scholar
  46. [46]
    Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos. Topology Appl., 117(3), 259–272 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    Huang, W., Ye, X.: Topological complexity, return times and weak disjointness. Ergodic Theory Dynam. Systems, 24(3), 825–846 (2004)MathSciNetzbMATHGoogle Scholar
  48. [48]
    Huang, W., Ye, X.: Dynamical systems disjoint from any minimal system. Trans. Amer. Math. Soc., 357(2), 669–694 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Huang, W., Ye, X.: A local variational relation and applications. Israel J. Math., 151, 237–279 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    Iwanik, A.: Independence and scrambled sets for chaotic mappings. In: The mathematical heritage of C. F. Gauss, World Sci. Publ., River Edge, NJ, 1991, 372–378CrossRefGoogle Scholar
  51. [51]
    Janková, K., Smítal, J.: A characterization of chaos. Bull. Austral. Math. Soc., 34(2), 283–292 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    Katznelson, Y., Weiss, B.: When all points are recurrent/generic. In: Ergodic theory and dynamical systems, I (College Park, Md., 1979–80), Progr. Math., Vol. 10, Birkhäuser Boston, Mass., 1981, 195–210CrossRefGoogle Scholar
  53. [53]
    Kerr, D., Li, H.: Independence in topological and C*-dynamics. Math. Ann., 338(4), 869–926 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    Kerr, D., Li, H.: Combinatorial independence and sofic entropy. Commun. Math. Stat., 1(2), 213–257 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    Keynes, H. B., Robertson, J. B.: Eigenvalue theorems in topological transformation groups. Trans. Amer. Math. Soc., 139, 359–369 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    Kolyada, S.: Li–Yorke sensitivity and other concepts of chaos. Ukrain. Mat. Zh., 56(8), 1043–1061 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    Kolyada, S., Snoha, L.: Some aspects of topological transitivity — a survey. In: Iteration theory (ECIT 94) (Opava), Grazer Math. Ber., Vol. 334, Karl-Franzens-Univ. Graz, Graz, 1997, 3–35Google Scholar
  58. [58]
    Li, J.: Chaos and entropy for interval maps. J. Dynam. Differential Equations, 23(2), 333–352 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  59. [59]
    Li, J.: Transitive points via Furstenberg family. Topology Appl., 158(16), 2221–2231 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  60. [60]
    Li, J.: Equivalent conditions of devaney chaos on the hyperspace. J. Univ. Sci. Technol. China, 44(2), 93–95 (2014)MathSciNetGoogle Scholar
  61. [61]
    Li, J.: Localization of mixing property via Furstenberg families. Discrete Contin. Dyn. Syst., 35(2), 725–740 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    Li, J., Oprocha, P.: On n-scrambled tuples and distributional chaos in a sequence. J. Difference Equ. Appl., 19(6), 927–941 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  63. [63]
    Li, J., Oprocha, P., Zhang, G.: On recurrence over subsets and weak mixing. Preprint (2013)Google Scholar
  64. [64]
    Li, J., Tu, S.: On proximality with Banach density one. J. Math. Anal. Appl., 416(1), 36–51 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  65. [65]
    Li, J., Tu, S., Ye, X.: Mean equicontinuity and mean sensitivity. Ergodic Theory Dynam. Systems, to appearGoogle Scholar
  66. [66]
    Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discrete Contin. Dyn. Syst., 35(3), 1059–1073 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    Li, R., Shi, Y.: Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. Abstr. Appl. Anal., Art. ID 769523, 10 pages (2014)Google Scholar
  68. [68]
    Li, S.: ω-chaos and topological entropy. Trans. Amer. Math. Soc., 339(1), 243–249 (1993)MathSciNetzbMATHGoogle Scholar
  69. [69]
    Li, T., Yorke, J. A.: Period three implies chaos. Amer. Math. Monthly, 82(10), 985–992 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  70. [70]
    Liao, G., Fan, Q.: Minimal subshifts which display Schweizer–Smítal chaos and have zero topological entropy. Sci. China Ser. A, 41(1), 33–38 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  71. [71]
    Liu, H., Liao, L., Wang, L.: Thickly syndetical sensitivity of topological dynamical system. Discrete Dyn. Nat. Soc., Art. ID 583431, 4 pages (2014)Google Scholar
  72. [72]
    Lorenz, E. N.: Deterministic nonperiodic flow. J. Atmospheric Sci., 20(2), 130–148 (1963)CrossRefGoogle Scholar
  73. [73]
    Mai, J.: Continuous maps with the whole space being a scrambled set. Chinese Sci. Bull., 42(19), 1603–1606 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  74. [74]
    Mai, J.: The structure of equicontinuous maps. Trans. Amer. Math. Soc., 355(10), 4125–4136 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  75. [75]
    Mai, J.: Devaney’s chaos implies existence of s-scrambled sets. Proc. Amer. Math. Soc., 132(9), 2761–2767 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  76. [76]
    Moothathu, T. K. S.: Stronger forms of sensitivity for dynamical systems. Nonlinearity, 20(9), 2115–2126 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  77. [77]
    Moothathu, T. K. S.: Syndetically proximal pairs. J. Math. Anal. Appl., 379(2), 656–663 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  78. [78]
    Moothathu, T. K. S., Oprocha, P.: Syndetic proximality and scrambled sets. Topol. Methods Nonlinear Anal., 41(2), 421–461 (2013)MathSciNetzbMATHGoogle Scholar
  79. [79]
    Mycielski, J.: Independent sets in topological algebras. Fund. Math., 55, 139–147 (1964)MathSciNetzbMATHGoogle Scholar
  80. [80]
    Oprocha, P.: Relations between distributional and Devaney chaos. Chaos, 16(3), 033112, 5 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    Oprocha, P.: Minimal systems and distributionally scrambled sets. Bull. Soc. Math. France, 140(3), 401–439 (2012)MathSciNetzbMATHGoogle Scholar
  82. [82]
    Oprocha, P., Zhang, G.: On local aspects of topological weak mixing in dimension one and beyond. Studia Math., 202(3), 261–288 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  83. [83]
    Oprocha, P., Zhang, G.: On weak product recurrence and synchronization of return times. Adv. Math., 244, 395–412 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  84. [84]
    Oprocha, P., Zhang, G.: On local aspects of topological weak mixing, sequence entropy and chaos. Ergodic Theory Dynam. Systems, 34(5), 1615–1639 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  85. [85]
    Oprocha, P., Zhang, G.: Topological aspects of dynamics of pairs, tuples and sets. In: Recent progress in general topology, III, Atlantis Press, Paris, 2014, 665–709CrossRefGoogle Scholar
  86. [86]
    Pikula, R.: On some notions of chaos in dimension zero. Colloq. Math., 107(2), 167–177 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  87. [87]
    Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals, 17(1), 99–104 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  88. [88]
    Ruelle, D.: Dynamical systems with turbulent behavior. In: Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), Lecture Notes in Phys., Vol. 80, Springer, Berlin, 1978, 341–360CrossRefGoogle Scholar
  89. [89]
    Ruelle, D., Takens, F.: On the nature of turbulence. Comm. Math. Phys., 20, 167–192 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  90. [90]
    Ruette, S.: Transitive sensitive subsystems for interval maps. Studia Math., 169(1), 81–104 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  91. [91]
    Scarpellini, B.: Stability properties of flows with pure point spectrum. J. London Math. Soc. (2), 26(3), 451–464 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  92. [92]
    Schweizer, B., Smítal, J.: Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc., 344(2), 737–754 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  93. [93]
    Shao, S.: Proximity and distality via Furstenberg families. Topology Appl., 153(12), 2055–2072 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  94. [94]
    Shao, S., Ye, X., Zhang, R.: Sensitivity and regionally proximal relation in minimal systems. Sci. China Ser. A, 51(6), 987–994 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  95. [95]
    Smítal, J.: A chaotic function with some extremal properties. Proc. Amer. Math. Soc., 87(1), 54–56 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  96. [96]
    Smítal, J.: Chaotic functions with zero topological entropy. Trans. Amer. Math. Soc., 297(1), 269–282 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    Smítal, J.: Topological entropy and distributional chaos. Real Anal. Exchange, (30th Summer Symposium Conference), 61–65 (2006)Google Scholar
  98. [98]
    Smítal, J., Stefánková, M.: Distributional chaos for triangular maps. Chaos Solitons Fractals, 21(5), 1125–1128 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    Snoha, L.: Generic chaos. Comment. Math. Univ. Carolin., 31(4), 793–810 (1990)MathSciNetzbMATHGoogle Scholar
  100. [100]
    Tan, F., Fu, H.: On distributional n-chaos. Acta Math. Sci. Ser. B Engl. Ed., 34(5), 1473–1480 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  101. [101]
    Tan, F., Xiong, J.: Chaos via Furstenberg family couple. Topology Appl., 156(3), 525–532 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  102. [102]
    Tan, F., Zhang, R.: On F-sensitive pairs. Acta Math. Sci. Ser. B Engl. Ed., 31(4), 1425–1435 (2011)MathSciNetzbMATHGoogle Scholar
  103. [103]
    Walters, P.: An introduction to ergodic theory. In: Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York, 1982Google Scholar
  104. [104]
    Wang, H., Xiong, J., Tan, F.: Furstenberg families and sensitivity. Discrete Dyn. Nat. Soc., Art. ID 649348, 12 pages (2010)Google Scholar
  105. [105]
    Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. In: Texts in Applied Mathematics, Vol 2, Springer-Verlag, New York, 1990Google Scholar
  106. [106]
    Xiong, J.: A chaotic map with topological entropy. Acta Math. Sci. English Ed., 6(4), 439–443 (1986)MathSciNetzbMATHGoogle Scholar
  107. [107]
    Xiong, J.: Chaos in a topologically transitive system. Sci. China Ser. A, 48(7), 929–939 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  108. [108]
    Xiong, J., LÜ, J., Tan, F.: Furstenberg family and chaos. Sci. China Ser. A, 50(9), 1325–1333 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  109. [109]
    Xiong, J., Yang, Z.: Chaos caused by a topologically mixing map. In: Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, Vol. 9, 550–572, World Sci. Publ., River Edge, NJ, 1991MathSciNetGoogle Scholar
  110. [110]
    Ye, X., Yu, T.: Sensitivity, proximal extension and higher order almost automorphy. Preprint (2014)Google Scholar
  111. [111]
    Ye, X., Zhang, R.: On sensitive sets in topological dynamics. Nonlinearity, 21(7), 1601–1620 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  112. [112]
    Yuan, D., LÜ, J.: Invariant scrambled sets in transitive systems. Adv. Math. (China), 38(3), 302–308 (2009)MathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsShantou UniversityShantouP. R. China
  2. 2.Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and School of MathematicsUniversity of Science and Technology of ChinaHefeiP. R. China

Personalised recommendations