[EN] Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800-6827, 2016). In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the -limit and the -limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.Wen Huang and Sergii Kolyada acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai. Sergii Kolyada also acknowledges the hospitality of the Max-Planck-Institute fur Mathematik (MPIM) in Bonn, the Departament de Matematica Aplicada of the Universitat Politecnica de Valencia, the partial support of Project MTM2013-47093-P, and the Department of Mathematics of the Chinese University of Hong Kong. We thank the referees for careful reading and constructive comments that have resulted in substantial improvements to this paper. Wen Huang was supported by NNSF of China (11225105, 11431012); Alfred Peris was supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P, and by GVA, Project PROMETEOII/2013/013; and Guohua Zhang was supported by NNSF of China (11671094).Huang, W.; Khilko, D.; Kolyada, S.; Peris Manguillot, A.; Zhang, G. (2018). Finite Intersection Property and Dynamical Compactness. Journal of Dynamics and Differential Equations. 30(3):1221-1245. https://doi.org/10.1007/s10884-017-9600-8S12211245303Akin, E.: Recurrence in topological dynamics. The University Series in Mathematics, Plenum Press, New York, Furstenberg families and Ellis actions (1997)Akin, E., Auslander, J., Berg, K.: When is a transitive map chaotic Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, pp. 25–40, de Gruyter, Berlin (1996)Akin, E., Glasner, E.: Residual properties and almost equicontinuity. J. Anal. Math. 84, 243–286 (2001)Akin, E., Kolyada, S.: Li–Yorke sensitivity. Nonlinearity 16(4), 1421–1433 (2003)Auslander, J.: Minimal flows and their extensions. North-Holland Mathematics Studies, vol. 153. North-Holland Publishing Co., Amsterdam, Notas de Matemática [Mathematical Notes], 122 (1988)Auslander, J., Yorke, J.A.: Interval maps, factors of maps, and chaos. Tôhoku Math. J. (2) 32(2), 177–188 (1980)Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167(1), 94–112 (1999)Blanchard, F., Huang, W.: Entropy sets, weakly mixing sets and entropy capacity. Discrete Contin. Dyn. Syst. 20(2), 275–311 (2008)de la Rosa, M., Read, C.: A hypercyclic operator whose direct sum T⊕T T ⊕ T is not hypercyclic. J. Oper. Theory 61(2), 369–380 (2009)Dowker, Y.N., Friedlander, F.G.: On limit sets in dynamical systems. Proc. Lond. Math. Soc. (3) 4, 168–176 (1954)Downarowicz, T.: Survey of odometers and Toeplitz flows. Algebraic and topological dynamics. Contemp. Math., vol. 385, Amer. Math. Soc., Providence, RI, pp. 7–37 (2005)Edwards, R.E.: Functional analysis. Dover Publications Inc, New York. Theory and applications. Corrected reprint of the 1965 original (1995)Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures. Princeton University Press, Princeton, NJ (1981)Furstenberg, H., Weiss, B.: Topological dynamics and combinatorial number theory. J. Anal. Math. 34(1978), 61–85 (1979)Glasner, E., Weiss, B.: Sensitive dependence on initial conditions. Nonlinearity 6(6), 1067–1075 (1993)Grosse-Erdmann, K.-G., Peris, A.: Weakly mixing operators on topological vector spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, vol. 104, no. 2, pp. 413–426 (2010)Grosse-Erdmann, K.-G., Peris-Manguillot, A.: Linear chaos, Universitext. Springer, London (2011)Guckenheimer, J.: Sensitive dependence to initial conditions for one-dimensional maps. Commun. Math. Phys. 70(2), 133–160 (1979)Halpern, J.D.: Bases in vector spaces and the axiom of choice. Proc. Am. Math. Soc. 17, 670–673 (1966)He, W.H., Zhou, Z.L.: A topologically mixing system whose measure center is a singleton. Acta Math. Sin. (Chin. Ser.) 45(5), 929–934 (2002)Huang, W., Khilko, D., Kolyada, S., Zhang, G.: Dynamical compactness and sensitivity. J. Differ. Equ. 260(9), 6800–6827 (2016)Huang, W., Kolyada, S., Zhang, G.: Analogues of Auslander–Yorke theorems for multi-sensitivity. Ergod. Theory Dyn. Syst. 22, 1–15 (2016). doi: 10.1017/etds.2016.48Huang, W., Ye, X.: Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topol. Appl. 117(3), 259–272 (2002)Kelley, J.L.: General topology. Graduate Texts in Mathematics, vol. 27. Springer, New York. Reprint of the 1955 edition [Van Nostrand, Toronto, ON] (1975)Kolyada, S., Snoha, L., Trofimchuk, S.: Noninvertible minimal maps. Fund. Math. 168(2), 141–163 (2001)Li, J.: Transitive points via Furstenberg family. Topol. Appl. 158(16), 2221–2231 (2011)Li, J., Ye, X.D.: Recent development of chaos theory in topological dynamics. Acta Math. Sin. (Engl. Ser.) 32(1), 83–114 (2016)Liu, H., Liao, L., Wang, L.: Thickly syndetical sensitivity of topological dynamical system. Discrete Dyn. Nat. Soc. (2014). Art. ID 583431, 4Moothathu, T.K.S.: Stronger forms of sensitivity for dynamical systems. Nonlinearity 20(9), 2115–2126 (2007)Mycielski, J.: Independent sets in topological algebras. Fund. Math. 55, 139–147 (1964)Oprocha, P., Zhang, G.: On local aspects of topological weak mixing in dimension one and beyond. Stud. Math. 202(3), 261–288 (2011)Oprocha, P., Zhang, G.: On local aspects of topological weak mixing, sequence entropy and chaos. Ergod. Theory Dyn. Syst. 34(5), 1615–1639 (2014)Petersen, K.E.: Disjointness and weak mixing of minimal sets. Proc. Am. Math. Soc. 24, 278–280 (1970)Read, C.J.: The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators. Isr. J. Math. 63(1), 1–40 (1988)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, pp. 341–360. Springer, Berlin (1978)Šarkovskiĭ, A.N.: Continuous mapping on the limit points of an iteration sequence. Ukrain. Mat. Ž. 18(5), 127–130 (1966)Weiss, B.: A survey of generic dynamics. Descriptive set theory and dynamical systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., vol. 277, pp. 273–291. Cambridge Univ. Press, Cambridge (2000
