# American Institute of Mathematical Sciences

January  2010, 9(1): 103-108. doi: 10.3934/cpaa.2010.9.103

## On the dynamics of flows on compact metric spaces

 1 Department of Mathematics, Kyungpook National University, Sankyuk-dong, Buk-gu, Daegu 702-701, South Korea 2 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Gwahak-ro, Yuseong-gu, Daejeon 305-701, South Korea

Received  February 2009 Revised  July 2009 Published  October 2009

In this paper, we consider a (generalized) envelope of flows on compact metric spaces. This partly generalizes the notion of envelope of maps in discrete geometry ([3]). We clarify a certain distinction between the flow geometry and the discrete one, which is explained by showing that any !-limit set for an envelope of flows is an empty set, whereas it is nonempty in general in discrete case.
Citation: Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103
 [1] Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671 [2] Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819 [3] Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787 [4] Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020065 [5] José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781 [6] Francisco Balibrea, J.L. García Guirao, J.I. Muñoz Casado. A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 983-994. doi: 10.3934/dcds.2002.8.983 [7] Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223 [8] José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010 [9] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019195 [10] Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 [11] Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487 [12] Jonathan Meddaugh, Brian E. Raines. The structure of limit sets for $\mathbb{Z}^d$ actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4765-4780. doi: 10.3934/dcds.2014.34.4765 [13] Wenyu Pan. Effective equidistribution of circles in the limit sets of Kleinian groups. Journal of Modern Dynamics, 2017, 11: 189-217. doi: 10.3934/jmd.2017009 [14] Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483 [15] Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393 [16] Krzysztof Barański. Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1015-1023. doi: 10.3934/dcds.2008.21.1015 [17] Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254 [18] Michael Winkler. Nontrivial ordered ω-limit sets in a linear degenerate parabolic equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 739-750. doi: 10.3934/dcds.2007.17.739 [19] Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059 [20] Ming Li, Shaobo Gan, Lan Wen. Robustly transitive singular sets via approach of an extended linear Poincaré flow. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 239-269. doi: 10.3934/dcds.2005.13.239

2018 Impact Factor: 0.925