zbMATH — the first resource for mathematics

Endomorphisms and automorphisms of the shift dynamical system. (English) Zbl 0182.56901
Let \((X(g),a)\) be the shift dynamical system, where the phase space \(X(g)\) of this system is the set of all bisequences over a finite symbol set \(\mathcal S\) with \(\mathrm{card }g>1\). The topology of \(X(g)\) is the product topology induced by the discrete topology of \(\mathcal S\). Let \(\Phi(\mathcal S)\) be the set of all continuous transformations of \(X(\mathcal S)\) into \(X(\mathcal S)\) which commute with \(\sigma\). One means of constructing such is to define an arbitrary mapping of blocks of specified length into single symbols and extending this mapping in a natural manner to infinite sequences. It has been shown by M. L. Curtis, the author and R. C. Lyndon that these mappings, composed with powers of the shift, constitute the entire class \(\Phi(\mathcal S)\). This result permits extensive analysis of the class \(\Phi(\mathcal S)\), the subclass \(E(\mathcal S)\) consisting of those members of \(\Phi(\mathcal S)\) which are onto, and the subgroup \(A(\mathcal S)\) of \(E(\mathcal S)\) consisting of those members which are one-to-one maps. Some of these results are the following.
Any finite group is isomorphic to some subgroup of \(A(\mathcal S)\) [Curtis, the author, Lyndon]. If \(\varphi\in E(\mathcal S)\) then there exists an integer \(K(\varphi)\) such that \(\mathrm{card}\ \varphi^{-1}(x) = K(\varphi)\) if \(x\) is bilaterally transitive (which is the case for almost all \(x)\) [A. M. Gleason and L. R. Welch].
If \(\varphi\in E(\mathcal S)\), and \(y\in\varphi^{-1}(x)\) then \(y\) is periodic if and only if \(x\) is periodic. The analogous result holds for almost periodicity, recurrence and transitivity. If \(\mathrm{card}\ \mathcal S\) is a prime and \(p\ge 2\), then there exists no continuous mapping \(\varphi\) such that \(\varphi^p =\sigma\) [L. R. Welch].
If \(\varphi\in E(\mathcal S)\), then the following statements are equivalent: (1) \(\varphi\) is an exactly \(\mu\)-to-one mapping of \(X(\mathcal S)\) onto \(X(\mathcal S)\); (2) \(\varphi\) is open; (3) \(\varphi\) has a cross-section; (4) for each \(x\in X(\mathcal S)\) any two distinct members of \(\varphi^{-1}(x)\) are distal [O. S. Rothaus].
Reviewer: G. A. Hedlund

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B10 Symbolic dynamics
54H20 Topological dynamics (MSC2010)
Full Text: DOI
[1] R. L. Adler, A. G. Konheim andM. H. McAndrew, Topological entropy,Trans. Amer. Math. Soc. 114 (1965), 309–319. · Zbl 0127.13102 · doi:10.1090/S0002-9947-1965-0175106-9
[2] Kiyoshi Aoki, On symbolic representation,Proc. Japan Acad. 30 (1954), 160–164. · Zbl 0058.37301 · doi:10.3792/pja/1195526143
[3] Emil Artin, Ein mechanisches System mit quasiergodischen Bahnen,Abh. Math. Sem. Univ. Hamburg 3 (1924), 170–175. · JFM 50.0677.11 · doi:10.1007/BF02954622
[4] G. D. Birkhoff,Dynamical Systems, Amer. Math. Soc. Colloq. Publ., Vol. 9, American Mathematical Society, Providence; first edition, 1927; revised edition, 1966. · JFM 53.0732.01
[5] G. D. Birkhoff, Sur le problème restreint des trois corps (Second mémoire),Ann. Scuola Nor. Sup. Pisa 5 (1936), 9–50. · JFM 62.0929.02
[6] J. D. Ferguson,Some Properties of Mappings on Sequence Spaces, Dissertation, Yale University, 1962. · Zbl 0111.33402
[7] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,Math. Systems Theory 1 (1967), 1–49. · Zbl 0146.28502 · doi:10.1007/BF01692494
[8] W. H. Gottschalk, Substitution minimal sets,Trans. Amer. Math. Soc. 109 (1963), 467–491. · Zbl 0121.18002 · doi:10.1090/S0002-9947-1963-0190915-6
[9] W. H. Gottschalk, Minimal sets: An introduction to topological dynamics,Bull. Amer. Math. Soc. 64 (1958), 336–351. · Zbl 0085.17401 · doi:10.1090/S0002-9904-1958-10223-2
[10] W. H. Gottschalk andG. A. Hedlund,Topological Dynamics, Amer. Math. Soc. Colloq. Publ., Vol. 36, American Mathematical Society, Providence, 1955. · Zbl 0067.15204
[11] W. H. Gottschalk andG. A. Hedlund, A characterization of the Morse minimal set,Proc. Amer. Math. Soc. 15 (1964), 70–74. · Zbl 0134.42203 · doi:10.1090/S0002-9939-1964-0158386-X
[12] Frank Hahn andYitzhak Katznelson, On the entropy of uniquely ergodic transformations.Trans. Amer. Math. Soc. 126 (1967), 335–360. · Zbl 0191.21502 · doi:10.1090/S0002-9947-1967-0207959-1
[13] Paul R. Halmos,Lectures on Ergodic Theory, Publ. Math. Soc. Japan, No. 3, Tokyo, 1956. · Zbl 0073.09302
[14] Paul R. Halmos,Entropy in Ergodic Theory, Univ. of Chicago Press, Chicago, 1959.
[15] G. A. Hedlund, Sturmian minimal sets,Amer. J. Math. 66 (1944), 605–620. · Zbl 0063.01982 · doi:10.2307/2371769
[16] G. A. Hedlund, Mappings on sequence spaces (Part I), Comm. Research Div. Technical Report No. 1, Princeton, N. J., Feb. 1961.
[17] G. A. Hedlund, Transformations commuting with the shift,Topological Dynamics (An International Symposium), Joseph Auslander and Walter H. Gottschalk (Editors), W. A. Benjamin, New York, 1968. · Zbl 0195.52702
[18] S. Kakutani, Ergodic Theory of Shift Transformations,Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability Theory II (1967), pp. 405–414. · Zbl 0217.38004
[19] E. A. Michael, Continuous selections II,Ann. of Math. 64 (1956), 562–580. · Zbl 0073.17702 · doi:10.2307/1969603
[20] Marston Morse, Recurrent geodesics on a surface of negative curvature,Trans. Amer. Math. Soc. 22 (1921), 84–100. · JFM 48.0786.06 · doi:10.1090/S0002-9947-1921-1501161-8
[21] Marston Morse,Symbolic Dynamics, Lectures by Marston Morse 1937–1938, Notes by Rufus Oldenburger, The Institute for Advanced Study, Princeton, N. J., 1966.
[22] Marston Morse andG. A. Hedlund, Symbolic dynamics,Amer. J. Math. 60 (1938), 815–866. · Zbl 0019.33502 · doi:10.2307/2371264
[23] Marston Morse andG. A. Hedlund, Symbolic dynamics II. Sturmian trajectories,Amer. J. Math. 62 (1940), 1–42. · Zbl 0022.34003 · doi:10.2307/2371431
[24] J. C. Oxtoby, Ergodic sets,Bull. Amer. Math. Soc. 58 (1952), 116–136. · Zbl 0046.11504 · doi:10.1090/S0002-9904-1952-09580-X
[25] William Parry, Symbolic dynamics and transformations of the unit interval,Trans. Amer. Math. Soc. 122 (1966), 368–378. · Zbl 0146.18604 · doi:10.1090/S0002-9947-1966-0197683-5
[26] William Reddy, Lifting expansive homeomorphisms to symbolic flows,Math. Systems Theory 2 (1968), 91–92. · Zbl 0157.29702 · doi:10.1007/BF01691348
[27] Stephen Smale, Diffeomorphisms with many periodic points,Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), edited by S. S. Cairns, Princeton University Press, Princeton, 1965, pp. 63–80.
[28] W. R. Utz, Almost periodic geodesics on manifolds of hyperbolic type,Duke Math. J. 18 (1951), 147–164. · Zbl 0044.17602 · doi:10.1215/S0012-7094-51-01812-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.