Varieties of mixing
HTML articles powered by AMS MathViewer
- by Ethan Akin and Jim Wiseman PDF
- Trans. Amer. Math. Soc. 372 (2019), 4359-4390 Request permission
Abstract:
We consider extensions of the notion of topological transitivity for a dynamical system $(X,f)$. In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each, there is a notion of mixing, defined by transitivity of the product system $(X \times X, f \times f)$. These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result in which a transitive system of each type either satisfies the corresponding mixing condition or else factors into an appropriate type of equicontinuous minimal system. The classical dichotomy result for minimal systems follows when it is shown that a minimal system is weak mixing if and only if it is vague mixing.References
- Ethan Akin, The general topology of dynamical systems, Graduate Studies in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1993. MR 1219737, DOI 10.1090/gsm/001
- Ethan Akin, On chain continuity, Discrete Contin. Dynam. Systems 2 (1996), no. 1, 111–120. MR 1367390, DOI 10.3934/dcds.1996.2.111
- E. Akin, Recurrence in topological dynamical systems: Furstenberg families and Ellis actions, Plenum Press, New York, 1997.
- Ethan Akin, Joseph Auslander, and Kenneth Berg, When is a transitive map chaotic?, Convergence in ergodic theory and probability (Columbus, OH, 1993) Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 25–40. MR 1412595
- Ethan Akin and Jeffrey D. Carlson, Conceptions of topological transitivity, Topology Appl. 159 (2012), no. 12, 2815–2830. MR 2942654, DOI 10.1016/j.topol.2012.04.016
- E. Akin and J. Wiseman, Chain recurrence for general spaces, arXiv:1707.0960v1 (2017).
- Joseph Auslander, Minimal flows and their extensions, North-Holland Mathematics Studies, vol. 153, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 122. MR 956049
- Joseph Auslander, Two folk theorems in topological dynamics, Eur. J. Math. 2 (2016), no. 2, 539–543. MR 3498998, DOI 10.1007/s40879-016-0097-1
- Joseph Auslander and Marianne Guerin, Regional proximality and the prolongation, Forum Math. 9 (1997), no. 6, 761–774. MR 1480556, DOI 10.1515/form.1997.9.761
- Joseph Auslander and James A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Math. J. (2) 32 (1980), no. 2, 177–188. MR 580273, DOI 10.2748/tmj/1178229634
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133, DOI 10.1090/cbms/038
- Robert Easton, Chain transitivity and the domain of influence of an invariant set, The structure of attractors in dynamical systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977) Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 95–102. MR 518550
- Albert Fathi and Pierre Pageault, Aubry-Mather theory for homeomorphisms, Ergodic Theory Dynam. Systems 35 (2015), no. 4, 1187–1207. MR 3345168, DOI 10.1017/etds.2013.107
- Eli Glasner and Benjamin Weiss, Sensitive dependence on initial conditions, Nonlinearity 6 (1993), no. 6, 1067–1075. MR 1251259, DOI 10.1088/0951-7715/6/6/014
- Sergiĭ Kolyada, L’ubomír Snoha, and Sergeĭ Trofimchuk, Noninvertible minimal maps, Fund. Math. 168 (2001), no. 2, 141–163. MR 1852739, DOI 10.4064/fm168-2-5
- Pierre Pageault, Conley barriers and their applications: chain-recurrence and Lyapunov functions, Topology Appl. 156 (2009), no. 15, 2426–2442. MR 2546945, DOI 10.1016/j.topol.2009.06.013
- David Richeson and Jim Wiseman, Chain recurrence rates and topological entropy, Topology Appl. 156 (2008), no. 2, 251–261. MR 2475112, DOI 10.1016/j.topol.2008.07.005
Additional Information
- Ethan Akin
- Affiliation: Department of Mathematics, The City College, 137 Street and Convent Avenue, New York, New York 10031
- MR Author ID: 24025
- Email: ethanakin@earthlink.net
- Jim Wiseman
- Affiliation: Department of Mathematics, Agnes Scott College, 141 East College Avenue, Decatur, Georgia 30030
- MR Author ID: 668909
- Email: jwiseman@agnesscott.edu
- Received by editor(s): October 13, 2017
- Received by editor(s) in revised form: June 29, 2018, and August 23, 2018
- Published electronically: November 5, 2018
- Additional Notes: The second author was supported by a grant from the Simons Foundation (282398, JW)
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4359-4390
- MSC (2010): Primary 37B20; Secondary 37B05, 37B35
- DOI: https://doi.org/10.1090/tran/7681
- MathSciNet review: 4009391