Počet záznamov: 1  

Topological size of scrambled sets

  1. NázovTopological size of scrambled sets
    Súbež.n.Topologická veľkosť chaotických množín
    Aut.údajeFrancois Blanchard, Wen Huang, Ľubomír Snoha
    Autor Blanchard Francois (34%)
    Spoluautori Huang Wen (33%)
    Snoha Ľubomír 1955- (33%) UMBFP10 - Katedra matematiky
    Zdroj.dok.Colloquium Mathematicum. Roč. 110, č. 2 (2008), s. 293-361. - Warszawa : Institute of Mathematics, Polish Academy of Sciences, 2008
    Kľúč.slová chaotická dvojica   chaotické množiny   Cantorova množina - Cantor set   Li-Yorkov chaos - Li-Yorke chaos   Mycielskeho množina   Bernsteinova množina   trojuholníkové zobrazenia   zobrazenia na grafe   topologická entropia - topological entropy   synchronizujúci podposun   scrambled pair   scrambled set   Cantor set   Mycielski set   Bernstein set   triangular maps   synchronising subshift   graph maps  
    Jazyk dok.angličtina
    KrajinaPoľsko
    AnotáciaA subset $S$ of a topological dynamical system $(X,f)$ containing at least two points is called a scrambled set if for any $x,y/in S$ with $x/neq y$ one has $/liminf_{n/to /infty} d(f^n(x), f^n(y)) = 0$ and $/limsup_{n/to /infty} d(f^n(x), f^n(y)) > 0,$ $d$ being the metric on $X$. The system $(X,f)$ is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about scrambled sets in the context of topological dynamics. There the assumption of Li-Yorke chaos, and also stronger ones like the existence of a residual scrambled set, or the fact that $X$ itself is a scrambled set (in these cases the system is called residually scrambled or completely scrambled respectively), are not so highly significant. But they still provide valuable information. First, the following question arises naturally: is it true in general that a Li-Yorke chaotic system has a Cantor scrambled set, at least when the phase space is compact? This question is not answered completely but the answer is known to be yes when the system is weakly mixing or Devaney chaotic or has positive entropy, all properties implying Li-Yorke chaos; we show that the same is true for symbolic systems and systems without asymptotic pairs, which may not be Li-Yorke chaotic. More generally, there are severe restrictions on Li-Yorke chaotic dynamical systems without a Cantor scrambled set, if they exist. A second set of questions concerns the size of scrambled sets inside the space $X$ itself. For which dynamical systems $(X,f)$ do there exist first category, or second category, or residual scrambled sets, or a scrambled set which is equal to the whole space $X$? While reviewing existing results, we give examples of systems on arc-wise connected continua in the plane having maximal scrambled sets with any prescribed cardinalities, in particular systems having at most finite or countable scrambled sets. We also give examples of Li-Yorke chaotic systems with at most first category scrambled sets. It is proved that minimal compact systems, graph maps and a large class of symbolic systems containing subshifts of finite type are never residually scrambled; assuming the Continuum Hypothesis, weakly mixing systems are shown to have second-category scrambled sets. Various examples of residually scrambled systems are constructed. It is shown that for any minimal distal system there exists a non-disjoint completely scrambled system. Finally various other questions are solved. For instance a completely scrambled system may have a factor without any scrambled set, and a triangular map may have a scrambled set with non-empty interior
    Kategória publikačnej činnosti ABA
    Číslo archívnej kópie9584
    Kategória ohlasu DOWNAROWICZ, T. Positive topological entropy implies chaos DC2. In Proceedings of the American mathematical society. ISSN 0002-9939, 2014, vol. 142, no. 1, pp. 137-149.
    BANKS, John - NGUYEN, Thi T. D. - OPROCHA, Piotr - STANLEY, Brett - TROTTA, Belinda. Dynamics of spacing shifts. In Discrete and continuous dynamical systems. ISSN 1078-0947, 2013, vol. 33, no. 9, pp. 4207-4232.
    MOOTHATHU, T. K. Subrahmonian - OPROCHA, Piotr. Syndetic proximality and scrambled sets. In Topological methods in nonlinear analysis. ISSN 1230-3429, 2013, vol. 41, no. 2, pp. 421-461.
    LI, Jian - OPROCHA, Piotr. On n-scrambled tuples and distributional chaos in a sequence. In Journal of difference equations and applications. ISSN 1023-6198, 2013, vol. 19, no. 6, pp. 927-941.
    FORYS, Magdalena - OPROCHA, Piotr - WILCZYNSKI, Pawel. Factor maps and invariant distributional chaos. In Journal of differential equations. ISSN 0022-0396, 2013, vol. 256, no. 2, pp. 475-502.
    DOWNAROWICZ, T. - LACROIX, Y. Topological entropy zero and asymptotic pairs. In Israel journal of mathematics. ISSN 0021-2172, 2012, vol. 189, no. 1, pp. 323-336.
    HUANG, Lin - WANG, Huoyun - WU, Hongying - YANG, WJ - LI, QS. A Remark on Invariant Scrambled Sets. In Progess in industrial and civil engineering. Zurich : Trans Tech Publications, 2012. Applied Mechanics and Materials, vol. 204-208. ISBN 978-3-03785-484-6, pp. 4776-4779.
    OPROCHA, Piotr. Coherent lists and chaotic sets. In Discrete and continuous dynamical systems. ISSN 1078-0947, 2011, vol. 31, no. 3, pp. 797-825.
    TAN FENG - ZHANG RUIFENG. On F-sensitive pairs. In Acta mathematica scientia. ISSN 0252-9602, 2011, vol. 31, no. 4, pp. 1425-1435.
    FU, Heman - XIONG, Jincheng - TAN, Feng. On distributionally chaotic and null systems. In Journal of mathematical analysis and applications. ISSN 0022-247X, 2011, vol. 375, no. 1, pp. 166-173.
    BRUIN, Henk - JIMENEZ LOPEZ, Victor. On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps. In Communications in mathematical physics. ISSN 0010-3616, 2010, vol. 299, no. 2, pp. 523-560.
    BALIBREA, F. - CARABALLO, T. - KLOEDEN, P. E. - VALERO, J. Recent developments in dynamical systems: three perspectives. In International journal of bifurcation and chaos. ISSN 0218-1274, 2010, vol. 20, no. 9, pp. 2591-2636.
    BALIBREA, Francisco - GUIRAO, Juan L. G. - OPROCHA, Piotr. On invariant epsilon-scrambled sets. In International journal of bifurcation and chaos. ISSN 0218-1274, 2010, vol. 20, no. 9, pp. 2925-2935.
    OPROCHA, Piotr. Families, filters and chaos. In Bulletin of the London mathematical society. ISSN 0024-6093, 2010, vol. 42, pp. 713-725.
    FU, Xin-Chu - CHEN, Zhan-He - GAO, Hongjun - LI, Chang-Pin - LIU, Zeng-Rong. Chaotic sets of continuous and discontinuous maps. In Nonlinear analysis-theory methods & applications. ISSN 0362-546X, 2010, vol. 72, no. 1, pp. 399-408.
    OPROCHA, Piotr. A note on distributional chaos with respect to a sequence. In Nonlinear analysis-theory methods & applications. ISSN 0362-546X, 2009, vol. 71, no. 11, pp. 5835-5839.
    OPROCHA, Piotr. Distributional chaos revisited. In Transactions of the American mathematical society. ISSN 0002-9947, 2009, vol. 361, no. 9, pp. 4901-4925.
    CIKLOVA-MLICHOVA, Michaela. Li-Yorke sensitive minimal maps II. In Nonlinearity. ISSN 0951-7715, 2009, vol. 22, no. 7, pp. 1569-1573.
    OPROCHA, Piotr. Invariant scrambled sets and distributional chaos. In Dynamical systems : an international journal. ISSN 1468-9367, 2009, vol. 24, no. 1, pp. 31-43.
    MOOTHATHU, T. K. Subrahmonian. Quantitative views of recurrence and proximality. In Nonlinearity. ISSN 0951-7715, 2008, vol. 21, no. 12, pp. 2981-2992.
    OPROCHA, Piotr - STEFANKOVA, Marta. Specification property and distributional chaos almost everywhere. In Proceedings of the American mathematical society. ISSN 0002-9939, 2008, vol. 136, no. 11, pp. 3931-3940.
    ASKRI, Ghassen - NAGHMOUCHI, Issam. Topological size of scrambled sets for local dendrite maps. In Topology and its applications. ISSN 0166-8641, 2014, vol. 164, pp. 95-104.
    FORYS, Magdalena - OPROCHA, Piotr - WILCZYNSKI, Pawel. Factor maps and invariant distributional chaos. In Journal of differential equations. ISSN 0022-0396, 2014, vol. 256, no. 2, pp. 475-502.
    TAN, Feng - FU, Heman. On distributional n-chaos. In Acta mathematica scientia. ISSN 0252-9602, 2014, vol. 34, no. 5, pp. 1473-1480.
    FALNIOWSKI, Fryderyk - KULCZYCKI, Marcin - KWIETNIAK, Dominik - LI, Jian. Two results on entropy, chaos and independence in symbolic dynamics. In Discrete and continuous dynamical systems - series B. ISSN 1531-3492, 2015, vol. 20, no. 10, pp. 3487-3505.
    LAMPART, Marek. Lebesgue measure of recurrent scrambled sets. In Springer proceedings in mathematics and statistics : 4th international workshop on nonlinear maps and their applications, NOMA 2013, Zaragoza, 3rd September - 4th September 2013. New York : Springer, 2015. ISBN 978-331912327-1, pp. 115-125.
    TAKACS, Michal. Generic chaos on graphs. In Journal of difference equations and applications. ISSN 1023-6198, 2016, vol. 22, no. 1, pp. 1-21.
    LI, Jian - YE, Xiang Dong. Recent development of chaos theory in topological dynamics. In Acta mathematica sinica - english series. ISSN 1439-8516, 2016, vol. 32, no. 1, pp. 83-114.
    LORANTY, Anna - PAWLAK, Ryszard J. On some sets of almost continuous functions which locally approximate a fixed function. In Real functions '15 : measure theory, real functions, genereal topology : 29th international summer conference on real functions theory, Niedzica, 06th-11th September 2015. ISSN 1210-3195, 2016, vol. 65, pp. 105-118.
    TAN, Feng. On an extension of Mycielski's theorem and invariant scrambled sets. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2016, vol. 36, pp. 632-648.
    LAMPART, Marek - OPROCHA, Piotr. Chaotic sub-dynamics in coupled logistic maps. In Physical D-nonlinear phenomena. ISSN 0167-2789, 2016, vol. 335, pp. 45-53.
    FORYS, Magdalena - HUANG, Wen - LI, Jian - OPROCHA, Piotr. Invariant scrambled sets, uniform rigidity and weak mixing. In Israel journal of mathematics. ISSN 0021-2172, 2016, vol. 211, no. 1, pp. 447-472.
    GARCIA-RAMOS, Felipe - JIN, Lei. Mean proximality and mean Li-Yorke chaos. In Proceedings of the American mathematical society. ISSN 0002-9939, 2017, vol. 145, no. 7, pp. 2959-2969.
    LI, Jian - OPROCHA, Piotr - YANG, Yini - ZENG, Tiaoying. On dynamics of graph maps with zero topological entropy. In Nonlinearity. ISSN 0951-7715, 2017, vol. 30, no. 12, pp. 4260-4276.
    NEUNHÄUSERE, J. Li-Yorke pairs of full Hausdoff dimension for some chaotic dynamical systems. In Mathematica Bohemica. ISSN 0862-7959, 2010, vol. 135, no. 3, pp. 279-289.
    FANG, Chun - HUANG, Wen - YI, Yingfei - ZHANG, Pengfei. Dimensions of stable sets and scrambled sets in positive finite entropy systems. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2012, vol. 32, pp. 599-628.
    SHIMOMURA, Takashi. Rank 2 proximal Cantor systems are residually scrambled. In Dynamical systems-an international journal. ISSN 1468-9367, 2018, vol. 33, no. 2, pp. 275-302.
    BORONSKI, Jan P. - KUPKA, Jiri - OPROCHA, Piotr. A mixing completely scrambled system exists. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2019, vol. 39, part 1, pp. 62-73.
    LI, Jian - LU, Jie - XIAO, Yuanfen. A dynamical version of the Kuratowski-Mycielski theorem and invariant chaotic sets. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2019, vol. 39, part. 11, pp. 3089-3110.
    LIN, Zijie - TAN, Feng. Generalized specification property and distributionally scrambled sets. In Journal of differential equations. ISSN 0022-0396, 2020, vol. 269, no. 7, pp. 5646-5660.
    TANG, Yan Jie - YIN, Jian Dong. Distributional chaos occurring on the set of proper positive upper banach density recurrent points of one-sided symbolic systems. In Acta mathematica sinica : english series. ISSN 1439-8516, 2020, vol. 36, no. 1, pp. 66-76.
    LI, Jian - LU, Jie - XIAO, Yuanfen. The Hausdorff dimension of multiply Xiong chaotic sets. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2020, vol. 40, no. 11, pp. 3056-3077.
    CHOTIBUT, Thiparat - FALNIOWSKI, Fryderyk - MISIUREWICZ, Michał - PILIOURAS, Georgios. The route to chaos in routing games : when is price of anarchy too optimistic? In Advances in neural information processing systems : 34th conference on neural information processing systems (NeurIPS 2020), virtual, 6th-12th December 2020. ISSN 1049-5258, 2020, vol. 33, pp. [1-12].
    CHEN, An - TIAN, Xueting. Distributional chaos in multifractal analysis, recurrence and transitivity. In Ergodic theory and dynamical systems. ISSN 0143-3857, 2021, vol. 41, no. 2, pp. 349-378.
    XIAO, Yuanfen. Mean li-yorke chaotic set along polynomial sequence with full hausdorff dimension for β-Transformation. In Discrete and continuous dynamical systems. ISSN 1078-0947, 2021, vol. 41, no. 2, pp. 525-536.
    GESCHKE, Stefan - GREBIK, Jan - MILLER, Benjamin D. Scrambled cantor sets. In Proceedings of the American mathematical society. ISSN 0002-9939, 2021, vol. 149, no. 10, pp. 4461-4468.
    ABDULSHAKOOR, Alqahtani Bushra M. - LIU, Weibin. Li-Yorke chaotic property of cookie-cutter systems. In AIMS mathematics. ISSN 2473-6988, 2022, vol. 7, no. 7, pp. 13192-13207.
    DENG, Liuchun - KHAN, M. Ali - RAJAN, Ashvin. Li-yorke chaos almost everywhere : on the pervasiveness of disjoint extremally scrambled sets. In Bulletin of the Australian mathematical society. ISSN 0004-9727, 2022, vol. 106, no. 1, pp. 132-143.
    DAGHAR, Aymen - NAGHMOUCHI, Issam. Entropy of induced maps of regular curves homeomorphisms. In Chaos, solitons and fractals. ISSN 0960-0779, 2022, vol. 157, art. no. 111988, pp. 1-6.
    TAN, Feng. Random dynamical systems with positive entropy imply second type of distributional chaos. In Journal of differential equations. ISSN 0022-0396, 2022, vol. 328, pp. 133-156.
    Katal.org.BB301 - Univerzitná knižnica Univerzity Mateja Bela v Banskej Bystrici
    Báza dátxpca - PUBLIKAČNÁ ČINNOSŤ
    nerozpoznaný

    nerozpoznaný

      Do košíka
Počet záznamov: 1  

  Tieto stránky využívajú súbory cookies, ktoré uľahčujú ich prezeranie. Ďalšie informácie o tom ako používame cookies.

Prijať všetkoNastavenieOdmietnuť všetko