zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. (English) Zbl 1172.37006
If $(E,f)$ is a dynamical system, then the hyperspace dynamical system $(\widehat{E},\widehat{f})$ is defined by $\widehat{f}(A):=f(A)$ on the collection $\widehat{E}$ of all subsets of $E$. The relation of different concepts of chaotic behaviour on $(E,f)$ and $(\widehat{E},\widehat{f})$ has been investigated in several papers. A map is said to depend sensitively on initial conditions (this property is briefly called sensitivity), if there is a $\delta>0$ such that for any $x\in E$ and any $\varepsilon>0$ there is a $y\in E$ with $d(y,x)<\varepsilon$ and an $n\in{\Bbb N}$ with $d(f\sp{n}(y),f\sp{n}(x))\ge \delta$. In this paper the authors introduce the notion of collective sensitivity. This means that there is a $\delta>0$ such that for finitely many $x\sb{1},x\sb{2},\dots ,x\sb{k}\in E$ and any $\varepsilon>0$ there are $y\sb{1},y\sb{2},\dots , y\sb{k}\in E$ with $d(y\sb{j},x\sb{j})<\varepsilon$ for all $j\in\{1,2,\dots ,k\}$ and there is an $n\in{\Bbb N}$ and a $u\in\{1,2,\dots ,k\}$ such that $d(f\sp{n}(y\sb{j}),f\sp{n}(x\sb{u}))\ge\delta$ for all $j\in\{1,2,\dots ,k\}$ or $d(f\sp{n}(x\sb{j}),f\sp{n}(y\sb{u}))\ge\delta$ for all $j\in\{1,2,\dots ,k\}$. It is proved that $(\widehat{E},\widehat{f})$ is sensitive if and only if $(E,f)$ is collectively sensitive. Here $\widehat{E}$ is endowed with the hit-or-miss topology. Moreover, also the conditions $({\Cal C},\widehat{f})$ is sensitive and $({\Cal F},\widehat{f})$ is sensitive are equivalent to $(\widehat{E},\widehat{f})$ is sensitive, where ${\Cal C}$ is the collection of all nonempty compact subsets of $E$ and ${\Cal F}$ is the collection of all nonempty finite subsets of $E$, both endowed with the Hausdorff metric (which is equivalent to the Vietoris topology in this case). The authors also prove that weak mixing implies collective sensitivity.

37B05Transformations and group actions with special properties
54B20Hyperspaces (general topology)
54H20Topological dynamics
Full Text: DOI
[1] Banks, J.: Chaos for induced hyperspace maps, Chaos solitons fractals 25, 681-685 (2005) · Zbl 1071.37012 · doi:10.1016/j.chaos.2004.11.089
[2] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P.: On devaney’s definition of chaos, Amer. math. Monthly 99, 332-334 (1992) · Zbl 0758.58019 · doi:10.2307/2324899
[3] Banks, J.: Regular periodic decompositions for topologically transitive maps, Ergodic theory dynam. Systems 17, 505-529 (1997) · Zbl 0921.54029 · doi:10.1017/S0143385797069885
[4] Bauer, W.; Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures, Monatsh. math. 79, 81-92 (1975) · Zbl 0314.54042 · doi:10.1007/BF01585664
[5] Beer, G.: Topologies on closed and closed convex sets, Math. appl. 268 (1993) · Zbl 0792.54008
[6] Beer, G.: On the fell topology, Set-valued anal. 1, 69-80 (1993) · Zbl 0810.54010 · doi:10.1007/BF01039292
[7] Bès, J.; Peris, A.: Hereditarily hypercyclic operators, J. funct. Anal. 167, 94-112 (1999) · Zbl 0941.47002 · doi:10.1006/jfan.1999.3437
[8] Block, L. S.; Coppel, W. A.: Dynamics in one dimension, Lecture notes in math. 1513 (1992) · Zbl 0746.58007
[9] Bowen, R.: Topological entropy and axiom A, Proc. sympos. Pure math. 14, 23-42 (1970) · Zbl 0207.54402
[10] Bowen, R.: Entropy for group endomorphisms and homogeneous spaces, Trans. amer. Math. soc. 14, 401-414 (1971) · Zbl 0212.29201 · doi:10.2307/1995565
[11] Devaney, R. L.: Chaotic dynamical system, (1989) · Zbl 0695.58002
[12] Engelking, R.: General topology, (1977) · Zbl 0373.54002
[13] Fedeli, A.: On chaotic set-valued discrete dynamical systems, Chaos solitons fractals 23, 1381-1384 (2005) · Zbl 1079.37021 · doi:10.1016/j.chaos.2004.06.039
[14] Fell, J. M. G.: A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. amer. Math. soc. 13, 472-476 (1962) · Zbl 0106.15801 · doi:10.2307/2034964
[15] Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. system theory 1, 1-49 (1967) · Zbl 0146.28502 · doi:10.1007/BF01692494
[16] Glasner, E.; Weiss, B.: Sensitive dependence on initial conditions, Nonlinearity 6, 1067-1075 (1993) · Zbl 0790.58025 · doi:10.1088/0951-7715/6/6/014
[17] Gu, R.; Guo, W.: On mixing property in set-valued discrete systems, Chaos solitons fractals 28, 747-754 (2006) · Zbl 1108.37004 · doi:10.1016/j.chaos.2005.04.004
[18] Illanes, A.; Jr., S. B. Nadler: Hyperspace: fundamentals and recent advances, (1999) · Zbl 0933.54009
[19] Kwietniak, D.; Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces, Chaos solitons fractals 33, 76-86 (2007) · Zbl 1152.37306 · doi:10.1016/j.chaos.2005.12.033
[20] Lechicki, A.; Levi, S.: Wijsman convergence in the hyperspace of a metric space, Bull. univ. Mat. ital. 7, No. 1-B, 439-452 (1987) · Zbl 0655.54007
[21] Liao, G.; Wang, L.; Zhang, Y.: Transitivity, mixing and chaos for a class of set-valued mappings, Sci. China ser. A 49, No. 1, 1-8 (2006) · Zbl 1193.37023 · doi:10.1007/s11425-004-5234-5
[22] Liao, G.; Ma, X.; Wang, L.: Individual chaos implies collective chaos for weakly mising discrete dynamical systems, Chaos solitons fractals 32, 604-608 (2007) · Zbl 1139.37010 · doi:10.1016/j.chaos.2005.11.002
[23] L. Liu, Y. Wang, G. Wei, Topological entropy of continuous functions on topological spaces, Chaos Solitons Fractals, in press (currently available at Elsevier’s website www.ScienceDirect.com)
[24] X. Ma, B. Hou, G. Liao, Chaos in hyperspace system, Chaos Solitons Fractals, in press (currently available at Elsevier’s website www.ScienceDirect.com)
[25] Matheron, G.: Random sets and integral geometry, (1975) · Zbl 0321.60009
[26] Michael, E.: Topologies on spaces of subsets, Trans. amer. Math. soc. 71, 152-182 (1951) · Zbl 0043.37902 · doi:10.2307/1990864
[27] Jr., S. B. Nadler: Hyperspaces of sets, Monogr. textb. Pure appl. Math. 49 (1978)
[28] Nguyen, H. T.; Wang, Y.; Wei, G.: On Choquet theorem for random upper semicontinuous functions, Internat. J. Approx. reason. 46, 3-16 (2007) · Zbl 1151.54013 · doi:10.1016/j.ijar.2006.12.004
[29] Nogura, T.; Shakhmatov, D.: When does the fell topology on a hyperspace of closed sets coincide with meet of the upper Kuratowski and the lower vietoris topology?, Topology appl. 70, 213-243 (1996) · Zbl 0848.54007 · doi:10.1016/0166-8641(95)00098-4
[30] Peña, J. S. C.; López, G. S.: Topological entropy for induced hyperspace maps, Chaos solitons fractals 28, 979-982 (2006) · Zbl 1097.54036 · doi:10.1016/j.chaos.2005.08.173
[31] Peris, A.: Set-valued discrete chaos, Chaos solitons fractals 26, 19-23 (2005) · Zbl 1079.37024 · doi:10.1016/j.chaos.2004.12.039
[32] Robinson, C.: Dynamical system: stability, symbolic dynamics, and chaos, (1999) · Zbl 0914.58021
[33] Rockafellar, R. T.; Wets, R. J. -B.: Variational systems, an introduction, Lecture notes in math. 1091, 1-54 (1984)
[34] Rockafellar, R. T.; Wets, R. J. -B.: Variational analysis, Grundlehren math. Wiss. 317, 144-147 (1998)
[35] Román-Flores, H.: A note on transitivity in set-valued discrete systems, Chaos solitons fractals 17, 99-104 (2003) · Zbl 1098.37008 · doi:10.1016/S0960-0779(02)00406-X
[36] Román-Flores, H.; Chalco-Cano, Y.: Robinson’s chaos in set-valued discrete systems, Chaos solitons fractals 25, 33-42 (2005) · Zbl 1071.37013
[37] Silverman, S.: On maps with dense orbits and the definition of chaos, Rocky mountain J. Math. 22, 353-375 (1992) · Zbl 0758.58024
[38] Stoyan, D.: Models and statistics, Internat. statist. Rev. 66, No. 1, 1-27 (1998) · Zbl 0906.60006
[39] Vellekoop, M.; Berglund, R.: On intervals, transitivity = chaos, Amer. math. Monthly 101, 353-355 (1994) · Zbl 0886.58033
[40] Viana, M.: Dynamics: A probabilistic and geometric perspective, Doc. math. Extra volume ICM 1, 557-578 (1998)
[41] Vietoris, L.: Bereiche zweiter ordnuang, Monatshefte für Mathematik und physik 33, 49-62 (1923)
[42] Walters, P.: An introduction to ergodic theory, Grad texts in math. 79 (1982) · Zbl 0475.28009
[43] Wang, Y.; Wei, G.: Conditions ensuring that hyperspace dynamical systems contain subsystems topologically (semi-)conjugate to symbolic dynamical systems, Chaos solitons fractals 36, 283-289 (2008) · Zbl 1135.54025 · doi:10.1016/j.chaos.2006.06.032
[44] Wang, Y.; Wei, G.: Embedding of topological dynamical systems into symbolic dynamical systems: A necessary and sufficient condition, Rep. math. Phys. 57, 457-461 (2006) · Zbl 1136.37008 · doi:10.1016/S0034-4877(06)80032-7
[45] Wang, Y.; Wei, G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems, Topology appl. 155, 56-68 (2007) · Zbl 1131.54024 · doi:10.1016/j.topol.2007.09.003
[46] Y. Wang, G. Wei, On constructing metrics of the hit-or-miss topology: Direct extensions, Pure Appl. Math. (2008), in press · Zbl 1199.54070
[47] Y. Wang, G. Wei, W.H. Campbell, S. Bourquin, A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology, Chaos Solitons Fractals, 2008, in press, doi:10.1016/j.chaos.2008.07.014 (currently available at Elsevier’s website www.ScienceDirect.com) · Zbl 1198.37026
[48] Watson, P. D.: On the limits of sequences of sets, Quart. J. Math. (2) 4, 1-3 (1953) · Zbl 0050.39201 · doi:10.1093/qmath/4.1.1
[49] G. Wei, Contributions to distributions of random sets on Polish spaces, PhD Thesis, New Mexico State University, Las Cruces, New Mexico, 1999
[50] Wei, G.; Wang, Y.: On metrization of the hit-or-miss topology using Alexandroff compactification, Internat. J. Approx. reason. 46, 47-64 (2007) · Zbl 1146.54004 · doi:10.1016/j.ijar.2006.12.007
[51] Wei, G.; Wang, Y.: Formulating stochastic convergence of random closed sets on locally compact separable metrizable spaces using metrics of the hit-or-miss topology, Internat. J. Intell. tech. Appl. stat. 1, 33-57 (2008)
[52] Zhang, G.; Zeng, F.; Liu, X.: Devaney’s chaotic on induced maps of hyperspace, Chaos solitons fractals 27, 471-475 (2006) · Zbl 1083.54026 · doi:10.1016/j.chaos.2005.03.053
[53] Zhou, Z. L.: Symbolic dynamical systems, (1997)