zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The transitivity of induced maps. (English) Zbl 1234.54017
Summary: For a metric continuum X, we consider the hyperspaces 2 X and C(X) of closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map f:XX we investigate the transitivity of the induced maps 2 f :2 X 2 X and C(f):C(X)C(X). Among other results, we show that if X is a dendrite or a continuum of type λ and f:XX is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map f:XX such that 2 f and C(f) are transitive.
MSC:
54B20Hyperspaces (general topology)
37B45Continua theory in dynamics
54F50Spaces of dimension 1; curves, dendrites
37B40Topological entropy
References:
[1]Acosta, G.; Eslami, P.; Oversteegen, L. G.: On open maps between dendrites, Houston J. Math. 33, No. 3, 753-770 (2007) · Zbl 1146.54006
[2]Banks, J.: Chaos for induced hyperspace maps, Chaos solitons fractals 25, No. 3, 681-685 (2005) · Zbl 1071.37012 · doi:10.1016/j.chaos.2004.11.089
[3]Barge, M.; Martin, J.: Chaos, periodicity, and snakelike continua, Trans. amer. Math. soc. 289, No. 1, 355-365 (1985) · Zbl 0559.58014 · doi:10.2307/1999705
[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]Bing, R. H.: Concerning hereditarily indecomposable continua, Pacific J. Math. 1, 43-51 (1951) · Zbl 0043.16803
[6]Block, L. S.; Coppel, W. A.: Dynamics in one dimension, Lecture notes in math. 1513 (1992) · Zbl 0746.58007
[7]Cánovas-Peña, J. S.; López, G. S.: Topological entropy for induced maps, Chaos solitons fractals 28, 979-982 (2006) · Zbl 1097.54036 · doi:10.1016/j.chaos.2005.08.173
[8]Charatonik, J. J.: On chaotic curves, Colloq. math. 41, 219-236 (1979) · Zbl 0447.54047
[9]Charatonik, J. J.: On chaotic dendrites, Period. math. Hungar. 38, No. 1 – 2, 19-29 (1999) · Zbl 0932.54034 · doi:10.1023/A:1004790813836
[10]Charatonik, W. J.; Dilks, A.: On self-homeomorphic spaces, Topology appl. 55, 215-238 (1994) · Zbl 0788.54040 · doi:10.1016/0166-8641(94)90038-8
[11]Cook, H.: Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. math. 60, 241-249 (1967) · Zbl 0158.41503
[12]Curtis, D. W.; Schori, R. M.: 2X and C(X) are homeomorphic to the Hilbert cube, Bull. amer. Math. soc. 80, 927-931 (1974) · Zbl 0302.54011 · doi:10.1090/S0002-9904-1974-13579-2
[13]Devaney, R. L.: An introduction to chaotic dynamical systems, (1989) · Zbl 0695.58002
[14]Efremova, L. S.; Makhrova, E. N.: The dynamics of monotone maps of dendrites, Mat. sb. 192, No. 6, 807-821 (2001) · Zbl 1008.37009 · doi:10.1070/SM2001v192n06ABEH000570
[15]Fedeli, A.: On chaotic set-valued discrete dynamical systems, Chaos solitons fractals 23, No. 4, 1381-1384 (2005) · Zbl 1079.37021 · doi:10.1016/j.chaos.2004.06.039
[16]Grispolakis, J.; Tymchatyn, E. D.: Irreducible continua with degenerate end-tranches and arcwise accessibility in hyperspaces, Fund. math. 110, 117-130 (1980) · Zbl 0378.54017
[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: Hyperspaces: fundamentals and recent advances, Monogr. textb. Pure appl. Math. 216 (1999) · Zbl 0933.54009
[19]Kuratowski, K.: Topology, vol. 2, (1968)
[20]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
[21]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
[22]X. Ma, B. Hou, G. Liao, Chaos in hyperspace system, Chaos Solitons Fractals, in press
[23]Méndez-Lango, H.: Some dynamical properties of mappings defined on knaster continua, Topology appl. 125, 419-428 (2002) · Zbl 1114.37300 · doi:10.1016/S0166-8641(02)00072-X
[24]Méndez-Lango, H.: The process of finding f ' for an entire function f has infinite topological entropy, Topology proc. 28, No. 2, 639-646 (2004) · Zbl 1082.37021
[25]Minc, P.; Transue, W. R. R.: A transitive map on [0,1] whose inverse limit is the pseudoarc, Proc. amer. Math. soc. 111, No. 4, 1165-1170 (1991) · Zbl 0767.54034 · doi:10.2307/2048584
[26]Jr., S. B. Nadler: Continuum theory: an introduction, Monogr. textb. Pure appl. Math. 158 (1992) · Zbl 0757.54009
[27]Peris, A.: Set-valued discrete chaos, Chaos solitons fractals 26, No. 1, 19-23 (2005) · Zbl 1079.37024 · doi:10.1016/j.chaos.2004.12.039
[28]Reńska, M.: Rigid hereditarily indecomposable continua, Topology appl. 126, 145-152 (2002) · Zbl 1017.54018 · doi:10.1016/S0166-8641(02)00075-5
[29]Román-Flores, H.: A note on transitivity in set-valued discrete systems, Chaos solitons fractals 17, No. 1, 99-104 (2003) · Zbl 1098.37008 · doi:10.1016/S0960-0779(02)00406-X
[30]Román-Flores, H.; Chalco-Cano, Y.: Robinson’s chaos in set-valued discrete systems, Chaos solitons fractals 25, No. 1, 33-42 (2005) · Zbl 1071.37013 · doi:10.1016/j.chaos.2004.11.006
[31]Špitalský, V.: Omega-limit sets in hereditarily locally connected continua, Topology appl. 155, 1237-1255 (2008) · Zbl 1184.37012 · doi:10.1016/j.topol.2008.03.005
[32]Walters, P.: An introduction to ergodic theory, Grad texts in math. 79 (1982) · Zbl 0475.28009
[33]Wang, Y.; Wei, G.: Characterizing mixing, weak mixing and transitivity on induced hyperspace dynamical systems, Topology appl. 155, 56-68 (2007) · Zbl 1131.54024 · doi:10.1016/j.topol.2007.09.003
[34]Zhang, G.; Zeng, F.; Liu, X.: Devaney’s chaotic on induced maps of hyperspaces, Chaos solitons fractals 27, No. 2, 471-475 (2006) · Zbl 1083.54026 · doi:10.1016/j.chaos.2005.03.053