Document Type : Research Paper

**Authors**

Department of pure mathematics, Ferdowsi university of Mashhad, Mashhad, Iran.

**Abstract**

In this paper, we introduce chaotic measure for discrete and continuous dynamical systems and study some properties of measure chaotic systems. Also relationship between chaotic measure, ergodic and expansive measures is investigated. Finally, we prove a new version of variational principle for chaotic measure.

**Keywords**

*Expansivity of ergodic measures with positive entropy*, ArXiv: 1110.5598 [math.DS], 2011.

[2] J. Banks, J. Brooks, G. Cairns and P. Stacey,

*On devaney’s definition of chaos*, Amer. Math. Monthly, 99 (1992), pp. 332-334.

[3] R. Bowen and P. Walters,

*Expansive one-parameter flows*, J. Differential Equations, 12 (1972), pp. 180-193.

[4] B. Cadre and P. Jacob,

*On pairwise sensitivity*, J. Math. Anal. Appl., 309 (2005), pp. 375-382.

[5] D. Carrasco-Olivera and C. A. Morales,

*Expansive measures for flows*, J. Differential Equations, 256 (2014), pp. 2246–2260.

[6] L. He, X. Yan and L. Wang,

*Weak-mixing implies sensitive dependence*, J. Math. Anal. Appl., 299 (2004), pp. 300-304.

[7] J. James, T. Koberda, k. Lindsey, P. Speh and C. E. Silva,

*Measurable Sensitivity*, Proc. Amer. Math. Soc., 136 (2008), pp. 3549-3559.

[8] L. Lam,

*Introduction to Nonlinear Physics*, Springer Verlag, New York, 1997.

[9] C.A. Morales,

*Measure-expansive Systems*, Preprint, IMPA, D083, (2011).

[10] W. Sun and E. Vargas,

*Entropy of flows*, Revisited. Bol. Soc. Brasil. Math. (N.S.), 30 (1999), pp. 315-333.

[11] W. Sun, T. Young and Y. Zhou,

*Topological entropies of equivalent smooth flows*, Trans. Amer. Math. Soc., 361 (2009), pp. 3071-3082.

[12] R.Thakur and R. Das,

*Devaney chaos and stronger forms of sensitivity on the product of semiflows*, Semigroup Forum, 98 (2019), pp. 631–644.

[13] P. Varandas,

*Entropy and Poincare recurrence from a geometrical viewpoint*, Nonlinearity, 22 (2009), pp. 2365-2375.

[14] P. Walters,

*An introduction to ergodic theory*, Springer Verlag, New York, 1982.

[15] A. Wilkinson,

*Smooth ergodic theory*, In Mathematics of complexity and dynamical systems. Springer, New York, 2012.

[16] L.S. Young,

*Entropy in dynamical systems*, Princeton University Press, (2003), pp. 313–328.