Abstract
In this Chapter examples of continuous and discrete flows will be discussed that are of particular interest. In Section 1 we study some ‘standard’ continuous and discrete flows on the circle 𝕋 and the torus 𝕋 2. Here the emphasis is on minimality. We also consider some skew product flows on tori.
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Notes
(6.1) Notes to Section 1
See [Hr], (8.14).
For other results on the existence of recurrent, nonperiodic points in a continuous flow on a compact surface, see A. Maĭer [1943] (the orientable case) and N.G. Markley [1969] (the non-orientable case). 5. See also [Hr] (26.19). 6. See M.C. Irwin [1980], 2.12. 7. This is adapted from H. Furstenberg [1981], Lemma 1.25. 8. For number theoretic consequences of this result, see H. Furstenberg [1981]. 9. See [Hr] (23.27) or, for a direct proof, P. Walters [1982], p. 14. 10. See W. Parry & P. Walters [1970].
(6.2) Notes to Section 2
This is by the characterization of the Cantor discontinuum by L.E.J. Brouwer; see R. Engelking [1977], Exercise 6.2A(c), or J.G. Hocking & G.S. Young [1961], Coro1lary 2–98.
References to the literature on symbolic flows up to 1940 can be found in M. Morse & G.A. Hedlund [1938], [1940]. Much information (including many references) can be found in G.A. Hedlund [1969] and, in relation with problems from ergodic theory, in M. Denker et al. [1976]. For the more recent applications of shift systems, see Note 8 below. Other applications include proofs of several results from combinatorial number theory as given in H. Furstenberg & B. Weiss [1978]; see also H. Furstenberg [1981] and, for an exposition for a general mathematical reader, J. De Vries [1 984]. Finally, sequences of symbols can be used for all sorts of book-keeping, e.g. the ‘kneading calculus’ for maps of the interval, introduced in J. Milnor & R. Thurston [1977] (this is related to symbolic dynamics); see also P. Collet & J.-P. Eckman [1980].
See B. Marcus [1985].
See G.A. Hedlund [1969].
In this context a shift system is often called a (noiseless) channel and a sofic system a ‘channel with infinite memory’.
Sofic systems were introduced in B. Weiss [1973] and developed further in, among others, E.M. Coven & M.E. Paul [1975; 1977]. In W. Krieger [1984/ 87] a method is given to obtain ‘canonically’ for a sofic system an ssft of which it is a factor. See also B. Marcus [1985]. For the classification of sofic systems, consult e.g. M. Boyle & W. Krieger [1988]. Many aspects of the characterizations of sofic systems in (5.10) can be traced back to the theory of finite-state automatons; for a first glance the relevant sections in G. Birkhoff & T.C. Bartee [1970] (Chapter 3 and Section 7 of Chapter 14) are quite instructive.
See G.A. Hedlund [1969], Theorems 5.5 and 5.13.
This method is due to Kakutani. The related results of (5.12) 4, 5 and V(6.1) 8 are well-known: see S. Kakutani [1967], W. Veech [1969], J.C. Martin [1971] and N.G. Markley [1974].
This is a topological version of an originally measure-theoretic construction of Chacón; see R.V. Chacón [1969]. That this topological version has the properties mentioned in (2.44) is implicit in N. Markley [1974] and was noticed again in A. Del Junco [1983]. Our presentation is based upon A. Del Junco [1982]. This flow is an example of a so-called Pod-flow; for an explanation of this term and the definition, see Note 17 in Iv(7.3). For further properties of Pod-flows, see Iv(3.49) 5, and V(6.4) 6. See also Note 20 below.
See [Wi], 12.2.5.
One of the reasons for their ubiquity in ergodic theory is that they form a class of ‘universal models’: according to the Jewett-Krieger Theorem, every ergodic m.p. transformation with finite entropy on a Lebesgue space can be realized as a uniquely ergodic minimal subshift. For the ergodic properties of the Morse system, see e.g. S. Kakutani [1967, 1972]; it is uniquely ergodic (for a proof, see M. Keane [1968]). For other special subshifts, see e.g. N.G. Markley [1974; 1975] and J.C. Martin [1976]. Other publications dealing directly or indirectly with the ergodic theory of shift systems are R. Adler & B. Weiss [1970], R. Adler & B. Marcus [1979] and, of course, R. Bowen [1975]. Of great interest are also F. Hann & Y. Katznelson [1967], K. Jacobs & M. Keane [1969], R.I. Jewett [1970], A. Del Junco, M. Keane, B. Kitchens, B. Marcus & L. Swanson [1981], W. Krieger [1972], W. Parry [1964] and W.A. Veech [1969].
Many questions remain: e.g., is every automorphism of a subshift the restriction of an automorphism of the full shift? See e.g. F. Rhodes [1988], also for the connection with the so-called ‘commuting block maps problem’. Or more generally (R.F. Williams): when does an automorphism of a finite subshift of a mixing ssft Λ extend to an automorphism of Λ? This problem is closely related to the classification problem of sofic systems: see M. Nasu [1988].
(6.3) Notes to Section 3
Here p stands for ‘right translation’. (Of course, as is Abelian, there is no difference between right and left translation. But much of this section will be repeated in a non-abelian context; see Section IV.5).
See H.H. Schaefer [1966], I.6.1, or (for topological groups) [Hr](8.3). For C c (ℝ), see also [Wi], Thm. 13.2.4.
See also S. Kono [1978].
In my opinion one cannot gain much insight from the fact that a flow < X, π > is isomorphic to a subflow of <C c (ℝ),ρ>, even though this shows that each π t is the restriction of a linear mapping.
(6.4) Notes to Section 4
See [Hr] (4.18)(f).
See G.A. Hedlund [1939] for an overview of the original approaches.
See H. Furstenberg [1973]. The unique ergodicity of more general horosphere flows was established in W.A. Veech [1977a]. For mixing properties of horocycle flows, see B. Marcus [1978], also for further references.
Formulation and proof of (4.7) are adapted from Chapter Ix of L. Auslander, L. Green & F. Hahn [1963]. It is related with a result from G.D. Mostov [1970].
Our proof is modelled after W.A. Veech [1975] as much as possible.
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© 1993 Springer Science+Business Media Dordrecht
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de Vries, J. (1993). Important Examples. In: Elements of Topological Dynamics. Mathematics and Its Applications, vol 257. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8171-4_3
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DOI: https://doi.org/10.1007/978-94-015-8171-4_3
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