Abstract
In this chapter we discuss a number of results from what might be called the ‘classical’ period in Topological Dynamics: the first half of this century—but some recent material is added. A central role is played by stability, in various forms. First, there is the theory of stability of invariant points and, more generally, invariant subsets (questions like: do they attract nearby orbits?). This topic will be almost neglected in this book. A second stability notion is that which can be roughly described by ‘recurrent behaviour’: after some time a given state is reached again (periodicity), or it is ‘almost’ reached infinitely often (various types of recurrence and almost periodicity). A different notion is transitivity: are there states from which every other state can be approached arbitrarily close? This notion turns out to have a close connection with notions of recurrence.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
(10.1) Notes to Section 1
One of the research papers in Topological Dynamics where the topology of the orbit space plays a role is O. Hajek [1971b].
See L.F. McAuley [1971].
In particular, we shall not study the local behaviour of a flow near an invariant point (cf. I (3.2)).
See e.g. K. Kuratowski [1966], II, §27, V.
A similar argument doesn’t work in continuous flows. See, however, Note 2 in (10.2) below.
(10.2) Notes to Section 2
The Poincaré-Bendixson Theorem (see I(1.5)) gives conditions for a flow in ℝ2 to have a limit set that is a periodic orbit. See also Note 10 in I(4.1).
(10.3) Notes to Section 3
A number of results about this problem is included in Chapter V of [Si]—but beware of the different terminology. In R.A. Knight [1981] it is claimed that if all points in a continuous flow on a locally compact space are recurrent, then all points are almost periodic. However, the proof is not correct. Moreover, the result is not true if the space is compact: see III(3.7)6. In R.A. Knight [1987] it is proved that in a compact discrete flow (X, f) with a dense orbit in which all points are positively recurrent there is a unique minimal set. I did not check all details of the proof, but the result seems to contradict the example on pp. 482–484 of V.V. Nemyckiy [1949] (cf. Note 6 in III (6.3) ahead).
See e.g., [Si], Th. 3.19, where it is ascribed to M.V. Bebutov [1940].
Modifications of this example occur at several places in the literature, serving various purposes. See e.g., [NS], Ex. 4.06 and 6.16 in Chapter V. The differential equations define a flow rather than a semiflow: see the final remark in Note 2 of I (4.1).
This notion already occurs in G.D. Birkhoff [1927].
In [NS], V.7.13 this trivial observation is ascribed to G.T. Tumarkin. This result implies that a compact minimal subset of a continuous flow in ℝn (or any other connected non-compact phase space) is nowhere dense. Using a result from dimension theory, it follows that a compact minimal subset of a continuous flow in ℝ n has dimension ≤n −1 (a result of H. Hilmy, 1937; see also [GH], 2.15); in addition, such a subset has the same dimension at every point (due to A.A. Markov [1931]; cf. [GH], 2.18). For discrete flows a similar result is not valid: see III(5.13). For a connected compact minimal discrete flow that is locally connected at some points but not locally connected at other points (due to G.B. Jones), see [GH], 14.18–14.24. A minimal set in a continuous flow is necessarily connected: by (3.7) it is the closure of a connected orbit. An example of Poincaré (cf. [NS], V.7.12; also [NS], V.8.15) shows that a compact minimal set need not be locally connected (see also III (5.13)). It is obvious that if X is locally connected and M is a minimal subet of X such that M is not locally connected, then M cannot be a periodic orbit and M has empty interior (for M cannot be clopen). Conversely, if X is a 2-manifold and M is a minimal subset that is not a periodic orbit and M has empty interior, then M is not locally connected (this follows easily from the local structure of the flow in any point of M, viz. the local parallelizability of the flow: see I (3.2). If the flows are sufficiently smooth then such minimal sets do not exist: see Note 10 in I(4.1).
This result goes back to Birkhoff. It appears in one form or another in many books. See e.g., [GH], 4.05,4.07 and 4.09; [El], 2.5; G.R. Sell [1971], VI.6 and [Br], 1.5.6. Also, [NS], 7.06 and 7.07 are equivalent with this theorem: see [NS], 7.09 (which is our (9.7)1). Compare also § 18 in [Si].
For classical results about minimal sets, see Note 10 in I(4.1); see also (9.6) 5. If the phase space of a minimal flow is connected and locally arcwise connected and admits a finite triangulation, then each orbit winds around the space in much the same way that occurs in the irrational flow on the torus; see the discussion following Theorem 2.1 in P.J. Kahn & A.W. Knapp [1968].
(10.4) Notes to Section 4
Most results in this section are standard: see e.g., P. Walters [1982] or M. Denker et al. [1976]. The notion of topologically weak mixing was introduced in H. Furstenberg [1967].
See H.B. Keynes & J.B. Robertson [1968].
For a more comprehensive discussion, see Section 4.5 in K. Petersen [1983].
(10.5) Notes to Section 5
The proof given here is taken from [El], 2.8. Related results can be found in [GH], 2.24 and [GH], 3.36. For the inheritance theorem for non-wandering points and recurrent points, see [GH] 7.21 and 7.04, respectively. The idea to use the notion of IP-sets in our proof of (9.12) is borrowed from H. Furstenberg [1981], Theorem 2.17. For the inheritance of recurrent points, see also [NS], V.4.12.
For the existence of a local cross-section, see Note 1 in I(4.3). Global cross-sections are usually obtained by glueing together local one’s. See e.g., H.A. Antosiewicz & J. Dugundji [1961], also O. Hajek [197 1b].
In a sense, suspensions are ubiquitous: every continuous flow without invariant points is a factor of the suspension of a shift system (cf. Section III.2), and the factor map is 1,1 between invariant residual sets: see H.B. Keynes & M. Sears [1979a]; see also H.B. Keynes, N.G. Markley & M. Sears [199?].
(10.6) Notes to Section 6
See Appendix (C.1). In the sequel, also Appendix (C.8) is used.
See [HR], Chapter 4.
Published by Poincaré in the Memoir on the stability of the solar system which won (January 21, 1889) the prize offered by King Oscar II of Sweden. It is published in Acta Math. 13 (1890), 1–270. The form we present here is essentially due to Caratheodory (however, both Poincaré and Caratheodory considered systems with continuous time).
Related with this result are two theorems by E. Hopf. Consider a (continuous or discrete) flow on a second countable space and assume that there is an invariant Borel measure (taking finite values on compact sets). Then either almost all points are pos. receding or almost all points are recurrent. In addition, almost all pos. receding points are neg. receding as well, and almost all pos. recurrent point are neg. recurrent. See [NS], Chapter VI, Section 4.
Other types of flows in which unique ergodicity is equivalent with minimality are e.g., affine transformations of compact metric Abelian groups (see (6.3) 5 for the definition). See F.J. Hahn & W. Parxy [1965] and A.H.M. Hoare & W. Parry [1966]. For ergodicity in more general flows, see Note 7 in I(4.2) and Notes 1, 8 and 11 in III (6.4). In this context the following problem (see P. Halmos [1956]) is interesting: can Haar measure on a non-compact locally compact topological group be ergodic under an automorphism of the group? According to N. Aoki [1985] the answer is ‘no’: ergodicity implies compactness.
(10.8) Notes to Section 8
More general situations are considered in R. Ellis [1965] and H.B. Keynes & D. Newton [1978]. Our presentation is adapted from S. Glasner & B. Weiss [1979]. For the case of a more general acting group than ℤ, see A. Fahti [1984].
See J.G. Hocking & G.S. Young [1961], Theorem 3–17.
See [HR] (8.14b).
See Theorem 19.4 in T.A. Chapman [1976].
See e.g., [Wi], Problem 201 of Section 13.2, or N. Bourbaki [1971], X.1.6, the Corollary to Theorem 2.
See [Du], IV.4.3.
This case was first established (with a different proof) in R. Peleg [1971].
See [HR] (23.26).
This is adapted from W. Parry [1969]. It is the topological counterpart of a theorem from H. Anzai [1951], where the notion of a skew-product transformation (of measure-preserving transformations) appeared for the first time. For a related result in a much more general context, see K.E. Petersen [1971].
See [Wi], Problem 119 in Section 6.7.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
de Vries, J. (1993). Continuous and Discrete Flows. In: Elements of Topological Dynamics. Mathematics and Its Applications, vol 257. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8171-4_2
Download citation
DOI: https://doi.org/10.1007/978-94-015-8171-4_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4274-3
Online ISBN: 978-94-015-8171-4
eBook Packages: Springer Book Archive