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**Generic distributional chaos and principal measure in linear dynamics.**
*(English)*
Zbl 1448.47019

B. Schweizer and J. Smítal introduced distributional chaos and principal measures in [Trans. Am. Math. Soc. 344, No. 2, 737–754 (1994; Zbl 0812.58062)]. A systematic investigation of distributional chaos for linear operators on Fréchet spaces was undertaken by Bernardes, Bonilla, Müller and Peris in [N. C. Bernardes jun. et al., J. Funct. Anal. 265, No. 9, 2143–2163 (2013; Zbl 1302.47014)]. Later on, several authors developed the theory of distributional chaos for \(C_0\)-semigroups of operators on Banach and Fréchet spaces (see the references in the article under review).

The authors deal in their paper with the notion of generic distributional chaos for \(C_0\)-semigroups of operators on Fréchet spaces. This means that the set of all distributionally chaotic pairs for a \(C_0\)-semigroup forms a residual set. Not every distributionally chaotic operator (or \(C_0\)-semigroup) has this property. The authors present some sufficient conditions for a \(C_0\)-semigroup of operators on a Fréchet space to be generically distributionally chaotic. Then they focus on a concrete example of a \(C_0\)-semigroup on the space of real-valued continuous functions on \([0,\infty[\), which is proved to be Devaney chaotic, topologically mixing, and generically distributionally chaotic with principal measure \(1\).

Furthermore, they consider distributionally chaotic dynamics and principal measures of product operators and product \(C_0\)-semigroups. They show that, under certain conditions, the product operator is generically distributionally chaotic if and only if there is a factor operator exhibiting generic distributional chaos. Every distributionally chaotic operator and every \(C_0\)-semigroup on a Banach space has principal measure 1. It is interesting that the authors show that there are distributionally chaotic operators whose principal measure is less than any given positive number. Moreover, such operators need not be hypercyclic and hence may not be Devaney chaotic.

The authors deal in their paper with the notion of generic distributional chaos for \(C_0\)-semigroups of operators on Fréchet spaces. This means that the set of all distributionally chaotic pairs for a \(C_0\)-semigroup forms a residual set. Not every distributionally chaotic operator (or \(C_0\)-semigroup) has this property. The authors present some sufficient conditions for a \(C_0\)-semigroup of operators on a Fréchet space to be generically distributionally chaotic. Then they focus on a concrete example of a \(C_0\)-semigroup on the space of real-valued continuous functions on \([0,\infty[\), which is proved to be Devaney chaotic, topologically mixing, and generically distributionally chaotic with principal measure \(1\).

Furthermore, they consider distributionally chaotic dynamics and principal measures of product operators and product \(C_0\)-semigroups. They show that, under certain conditions, the product operator is generically distributionally chaotic if and only if there is a factor operator exhibiting generic distributional chaos. Every distributionally chaotic operator and every \(C_0\)-semigroup on a Banach space has principal measure 1. It is interesting that the authors show that there are distributionally chaotic operators whose principal measure is less than any given positive number. Moreover, such operators need not be hypercyclic and hence may not be Devaney chaotic.

Reviewer: José Bonet (Valencia)

### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |

47D06 | One-parameter semigroups and linear evolution equations |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

46A04 | Locally convex Fréchet spaces and (DF)-spaces |

### Keywords:

generic distributional chaos; \(C_0\)-semigroups of operators; Fréchet spaces; principal measures
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\textit{Z. Yin} and \textit{Q. Yang}, Ann. Pol. Math. 118, No. 1, 71--94 (2016; Zbl 1448.47019)

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### References:

[1] | [1]A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators, Comm. Pure Appl. Anal. 12 (2013), 2069–2082. · Zbl 1287.47007 |

[2] | [2]X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations, Abstr. Appl. Anal. 2012 (2012), 457019. · Zbl 1256.37027 |

[3] | [3]X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sep’ulveda, Distributional chaos for the forward and backward control traffic model, Linear Algebra Appl. 479 (2015), 202–215. · Zbl 1330.47093 |

[4] | [4]X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Differential Equations Appl. 18 (2012), 751–761. · Zbl 1248.47011 |

[5] | [5]S. Bartoll, F. Mart’ınez-Gim’enez and A. Peris, Operators with the specification property, J. Math. Anal. Appl. 436 (2016), 478–488. · Zbl 1360.37024 |

[6] | [6]F. Bayart and T. Berm’udez, Dynamics of Linear Operators, Cambridge Univ. Press, Cambridge, 2009. |

[7] | [7]F. Bayart and Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dynam. Systems 35 (2015), 691–709. · Zbl 1355.37035 |

[8] | [8]T. Berm’udez, A. Bonilla, F. Mart’ınez-Gim’enez and A. Peris, Li–Yorke and distributionally chaotic operators, J. Math. Anal. Appl. 373 (2011), 83–93. Generic distributional chaos and principal measure93 · Zbl 1214.47012 |

[9] | [9]N. C. Bernardes, Jr., A. Bonilla, V. M”uller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal. 265 (2013), 2143–2163. · Zbl 1302.47014 |

[10] | [10]N. C. Bernardes, Jr., A. Bonilla, V. M”uller and A. Peris, Li–Yorke chaos in linear dynamics, Ergodic Theory Dynam. Systems 35 (2015), 1723–1745. · Zbl 1352.37100 |

[11] | [11]J. Bonet, A problem on the structure of Fr’echet spaces, RACSAM Rev. R. Acad. A 104 (2010), 427–434. · Zbl 1262.46001 |

[12] | [12]J. A. Conejero, M. Kosti’c, P. J. Miana and M. Murillo-Arcila, Distributionally chaotic families of operators on Fr’echet spaces, Comm. Pure Appl. Anal. 15 (2016), 1915–1939. · Zbl 1394.47013 |

[13] | [13]J. A. Conejero, F. Rodenas and M. Trujillo, Chaos for the hyperbolic bioheat equation, Discrete Contin. Dynam. Systems 35 (2015), 653–668. · Zbl 1330.47099 |

[14] | [14]R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., AddisonWesley, Redwood City, CA, 1989. · Zbl 0695.58002 |

[15] | [15]G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229–269. · Zbl 0732.47016 |

[16] | [16]K. G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011. · Zbl 1246.47004 |

[17] | [17]B. Z. Hou, P. Y. Cui and Y. Cao, Chaos for Cowen–Douglas operators, Proc. Amer. Math. Soc. 138 (2010), 929–936. |

[18] | [18]B. Z. Hou, G. Tian and S. Zhu, Approximation of chaotic operators, J. Operator Theory 67 (2012), 469–493. · Zbl 1261.47016 |

[19] | [19]E. M. Mangino and M. Murillo-Arcila, Frequently hypercyclic translation semigroups, Studia Math. 227 (2015), 219–238. · Zbl 1341.47007 |

[20] | [20]F. Mart’ınez-Gim’enez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl. 351 (2009), 607–615. |

[21] | [21]F. Mart’ınez-Gim’enez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets, Math. Z. 274 (2013), 603–612. |

[22] | [22]P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A 39 (2006), 14559–14565. · Zbl 1107.81026 |

[23] | [23]P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc. 361 (2009), 4901–4925. · Zbl 1179.37017 |

[24] | [24]B. Schweizer, A. Sklar and J. Sm’ıtal, Distributional (and other) chaos and its measurement, Real Anal. Exchange 26 (2000), 495–524. |

[25] | [25]B. Schweizer and J. Sm’ıtal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), 737–754. · Zbl 0812.58062 |

[26] | [26]F. Takeo, Chaos and hypercyclicity for solution semigroups to some partial differential equations, Nonlinear Anal. 63 (2005), 1943–1953. |

[27] | [27]X. X. Wu, Maximal distributional chaos of weighted shift operators on K”othe sequence spaces, Czechoslovak Math. J. 64 (2014), 105–114. · Zbl 1340.37058 |

[28] | [28]X. X. Wu, G. R. Chen and P. Y. Zhu, Invariance of chaos from backward shift on the K”othe sequence space, Nonlinearity 27 (2014), 271–288. · Zbl 1291.54046 |

[29] | [29]X. X. Wu, P. Oprocha and G. R. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity 29 (2016), 1942–1972. · Zbl 1364.37023 |

[30] | [30]X. X. Wu and P. Y. Zhu, The principal measure of a quantum harmonic oscillator, J. Phys. A 44 (2011), 505101, 6 pp. · Zbl 1238.81129 |

[31] | [31]X. X. Wu and P. Y. Zhu, Chaos in the weighted Biebutov systems, Int. J. Bifur. Chaos 23 (2013), no. 8, 1350133, 9 pp. · Zbl 1275.37019 |

[32] | [32]Z. B. Yin and Q. G. Yang, Distributionally scrambled set for an annihilation operator, Int. J. Bifur. Chaos 25 (2015), no. 13, 1550178, 13 pp. · Zbl 1330.81103 |

[33] | [33]Y. H. Zhou, Distributional chaos for flows, Czechoslovak Math. J. 63 (2013), 475– 480. 94Z. B. Yin and Q. G. Yang |

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