• Previous Article
    Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation
  • DCDS Home
  • This Issue
  • Next Article
    Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$
January  1999, 5(1): 83-92. doi: 10.3934/dcds.1999.5.83

Topological mapping properties defined by digraphs

1. 

Department of Mathematics, La Trobe University Bundoora, Australia 3083, Australia

Received  February 1998 Revised  July 1998 Published  October 1998

Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. Using finite directed graphs, these definitions are generalized to obtain topological mapping properties. The extent to which these mapping properties are logically distinct is examined. There are three distinct properties which entail "interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does notappear in the literature but turns out to be close to weak mixing in a sense to be discussed. The remaining properties comprise a countably infinite collection of distinct properties entailing somewhat less interesting dynamics and including non-wandering.
Citation: John Banks. Topological mapping properties defined by digraphs. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83
[1]

John Banks, Brett Stanley. A note on equivalent definitions of topological transitivity. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1293-1296. doi: 10.3934/dcds.2013.33.1293

[2]

Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175

[3]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

[4]

Yang Cao, Song Shao. Topological mild mixing of all orders along polynomials. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1163-1184. doi: 10.3934/dcds.2021150

[5]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993

[6]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195

[7]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[8]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

[9]

Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168

[10]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079

[11]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427

[12]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35

[13]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321

[14]

Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929

[15]

Sergio Muñoz. Robust transitivity of maps of the real line. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1163-1177. doi: 10.3934/dcds.2015.35.1163

[16]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

[17]

Gernot Greschonig. Regularity of topological cocycles of a class of non-isometric minimal homeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4305-4321. doi: 10.3934/dcds.2013.33.4305

[18]

Ming-Chia Li, Ming-Jiea Lyu. Topological conjugacy for Lipschitz perturbations of non-autonomous systems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5011-5024. doi: 10.3934/dcds.2016017

[19]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064

[20]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (179)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]