Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy

We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page

Induced dynamics on the hyperspaces

[EN]  In this paper, we study the dynamics induced by finite commutative relation on the hyperspaces. We prove that the dynamics induced on the hyperspace by a non-trivial commutative family of continuous self maps cannot be transitive and hence cannot exhibit higher degrees of mixing. We also prove that the dynamics induced on the hyperspace by such a collection cannot have dense set of periodic points. We also give example to show that the induced dynamics in this case may or may not be sensitive.Sharma, P. (2016). Induced dynamics on the hyperspaces. Applied General Topology. 17(2):93-104. doi:10.4995/agt.2016.4154.SWORD93104172Banks, J. (2005). Chaos for induced hyperspace maps. Chaos, Solitons & Fractals, 25(3), 681-685. doi:10.1016/j.chaos.2004.11.089Michael, E. (1951). Topologies on spaces of subsets. Transactions of the American Mathematical Society, 71(1), 152-152. doi:10.1090/s0002-9947-1951-0042109-4Harper, M., & Hunter, J. (2010). Introduction to new series. Northern Scotland, 1(1), 1-2. doi:10.3366/nor.2010.0001Sharma, P., & Nagar, A. (2010). Inducing sensitivity on hyperspaces. Topology and its Applications, 157(13), 2052-2058. doi:10.1016/j.topol.2010.05.002Román-Flores, H. (2003). A note on transitivity in set-valued discrete systems. Chaos, Solitons & Fractals, 17(1), 99-104. doi:10.1016/s0960-0779(02)00406-

Set-Valued Chaos in Linear Dynamics

[EN] We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator on a topological vector space X, and the natural hyperspace extensions and of T to the spaces of compact subsets of X and of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, and . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681-685, 2005) and Peris (Chaos Solitons Fractals 26(1):19-23, 2005).The first author was partially supported by CNPq (Brazil) and by the EBW+ Project (Erasmus Mundus Programme). The second and third authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second author was partially supported by GVA, Project PROMETEOII/2013/013.Bernardes, NCJ.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2017). Set-Valued Chaos in Linear Dynamics. Integral Equations and Operator Theory. 88(4):451-463. https://doi.org/10.1007/s00020-017-2394-6S451463884Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals 25(3), 681–685 (2005)Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79, 81–92 (1975)Bayart, F., Matheron, É.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250(2), 426–441 (2007)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li-Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35(6), 1723–1745 (2015)Bernardes Jr., N.C., Vermersch, R.M.: Hyperspace dynamics of generic maps of the Cantor space. Can. J. Math. 67(2), 330–349 (2015)Bès, J., Menet, Q., Peris, A., Puig, Y.: Strong transitivity properties for operators. Preprint ( arXiv:1703.03724 )Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167(1), 94–112 (1999)Bonet, J., Frerick, L., Peris, A., Wengenroth, J.: Transitive and hypercyclic operators on locally convex spaces. Bull. Lond. Math. Soc. 37, 254–264 (2005)de la Rosa, M., Read, C.: A hypercyclic operator whose direct sum $$T \oplus T$$ T ⊕ T is not hypercyclic. J. Oper. Theory 61(2), 369–380 (2009)Fedeli, A.: On chaotic set-valued discrete dynamical systems. Chaos Solitons Fractals 23(4), 1381–1384 (2005)Fu, H., Xing, Z.: Mixing properties of set-valued maps on hyperspaces via Furstenberg families. Chaos Solitons Fractals 45(4), 439–443 (2012)Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)Grivaux, S.: Hypercyclic operators, mixing operators, and the bounded steps problem. J. Oper. Theory 54, 147–168 (2005)Grosse-Erdmann, K.-G., Peris, A.: Weakly mixing operators on topological vector spaces. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104(2), 413–426 (2010)Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Universitext, Springer-Verlag, Berlin (2011)Guirao, J.L.G., Kwietniak, D., Lampart, M., Oprocha, P., Peris, A.: Chaos on hyperspaces. Nonlinear Anal. 71(1–2), 1–8 (2009)Hernández, P., King, J., Méndez, H.: Compact sets with dense orbit in $$2^X$$ 2 X . Topol. Proc. 40, 319–330 (2012)Herzog, G., Lemmert, R.: On universal subsets of Banach spaces. Math. Z. 229(4), 615–619 (1998)Hou, B., Tian, G., Shi, L.: Some dynamical properties for linear operators. Ill. J. Math. 53(3), 857–864 (2009)Hou, B., Tian, G., Zhu, S.: Approximation of chaotic operators. J. Oper. Theory 67(2), 469–493 (2012)Illanes, A., Nadler Jr., S.B.: Hyperspaces: Fundamentals and Recent Advances. Marcel Dekker Inc, New York (1999)Kupka, J.: On Devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst. 177(1), 34–44 (2011)Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33(1), 76–86 (2007)Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discrete Contin. Dyn. Syst. 35(3), 1059–1073 (2015)Liao, G., Wang, L., Zhang, Y.: Transitivity, mixing and chaos for a class of set-valued mappings. Sci. China Ser. A Math. 49(1), 1–8 (2006)Liu, H., Shi, E., Liao, G.: Sensitivity of set-valued discrete systems. Nonlinear Anal. 71(12), 6122–6125 (2009)Liu, H., Lei, F., Wang, L.: Li-Yorke sensitivity of set-valued discrete systems, J. Appl. Math. (2013). Article number: 260856Peris, A.: Set-valued discrete chaos. Chaos Solitons Fractals 26(1), 19–23 (2005)Peris, A., Saldivia, L.: Syndetically hypercyclic operators. Integral Equ. Oper. Theory 51, 275–281 (2005)Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals 17(1), 99–104 (2003)Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)Salas, H.: A hypercyclic operator whose adjoint is also hypercyclic. Proc. Am. Math. Soc. 112(3), 765–770 (1991)Shkarin, S.: Hypercyclic operators on topological vector spaces. J. Lond. Math. Soc. 86, 195–213 (2012)Wang, Y., Wei, G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Topol. Appl. 155(1), 56–68 (2007)Wang, Y., Wei, G., Campbell, W.H.: Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. Topol. Appl. 156(4), 803–811 (2009)Wu, Y., Xue, X., Ji, D.: Linear transitivity on compact connected hyperspace dynamics. Dyn. Syst. Appl. 21(4), 523–534 (2012)Wu, X., Wang, J., Chen, G.: F-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015