Bulletin of the American Mathematical Society

MathSciNet review: 1566980
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: J. de Vries
Title: Topological transformation groups 1: A categorical approach
Additional book information: Mathematical Centre Tracts, no. 65, Mathematisch Centrum, Amsterdam, 1975, v + 249 pp.

Joseph Auslander, Endomorphisms of minimal sets, Duke Math. J. 30 (1963), 605–614. MR 155311

L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, Princeton, N.J., 1963. MR 29 #4841.

L. Auslander, F. Hahn and L. Markus, Minimal flows on nilmanifolds, Flows on Homogeneous Spaces (Auslander, Green and Hahn, authors), Princeton Univ. Press, Princeton, N.J., 1963 (above).

Nam P. Bhatia, Dynamical systems, Mathematical systems theory and economics, I, II (Proc. Internat. Summer School, Varenna, 1967) Lecture Notes in Operations Research and Mathematical Economics, Vols. 11, vol. 12, Springer, Berlin, 1969, pp. 1–9. MR 0324143

G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.

G. Birkhoff, Quelques théorèmes sur le mouvement des systemes dynamiques, Bull. Soc. Math. France 40 (1912), 1-19.

George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095

G. D. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly 49 (1942), 222–226. MR 6619, DOI 10.2307/2303229

Harald Bohr, Almost Periodic Functions, Chelsea Publishing Co., New York, N.Y., 1947. MR 0020163

Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144

R. W. Brockett, Lie algebras and Lie groups in control theory, Harvard Univ. (preprint).

M. L. Cartwright, From non-linear oscillations to topological dynamics, J. London Math. Soc. 39 (1964), 193–201. MR 160988, DOI 10.1112/jlms/s1-39.1.193

Eduardo J. Dubuc, Kan extensions in enriched category theory, Lecture Notes in Mathematics, Vol. 145, Springer-Verlag, Berlin-New York, 1970. MR 0280560

Robert Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401–405. MR 101283

Robert Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272–281. MR 123636, DOI 10.1090/S0002-9947-1960-0123636-3

Robert Ellis, Universal minimal sets, Proc. Amer. Math. Soc. 11 (1960), 540–543. MR 117716, DOI 10.1090/S0002-9939-1960-0117716-1

Robert Ellis, The construction of minimal discrete flows, Amer. J. Math. 87 (1965), 564–574. MR 185589, DOI 10.2307/2373063

Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561

Robert Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258–271. MR 123635, DOI 10.1090/S0002-9947-1960-0123635-1

A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799

E. E. Floyd, A nonhomogeneous minimal set, Bull. Amer. Math. Soc. 55 (1949), 957–960. MR 33535, DOI 10.1090/S0002-9904-1949-09318-7

W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336–351. MR 100048, DOI 10.1090/S0002-9904-1958-10223-2

W. H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc. 109 (1963), 467–491. MR 190915, DOI 10.1090/S0002-9947-1963-0190915-6

W. H. Gottschalk, A survey of minimal sets, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 1, 53–60. MR 166771

Walter H. Gottschalk, Bibliography for topological dynamics, 3rd ed., Wesleyan University, Department of Mathematics, Middletown, Conn., 1967. MR 0224079

Walter Gottschalk, Some general dynamical notions, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 120–125. MR 0407821

Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological dynamics, American Mathematical Society Colloquium Publications, Vol. 36, American Mathematical Society, Providence, R.I., 1955. MR 0074810

Otomar Hájek, Dynamical systems in the plane, Academic Press, London-New York, 1968. MR 0240418

John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144

Harvey B. Keynes, On the proximal relation being closed, Proc. Amer. Math. Soc. 18 (1967), 518–522. MR 212780, DOI 10.1090/S0002-9939-1967-0212780-X

Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798

E. G. Manes, A characterization of dual spaces, J. Functional Analysis 17 (1974), 292–295. MR 0383048, DOI 10.1016/0022-1236(74)90042-1

Ernest G. Manes, Algebraic theories, Graduate Texts in Mathematics, No. 26, Springer-Verlag, New York-Heidelberg, 1976. MR 0419557

L. Markus, Lectures in differentiable dynamics, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 3, American Mathematical Society, Providence, R.I., 1971. MR 0309152

Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787

Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104

V. V. Nemyckiĭ and V. V. Stepanov, Kačestvennaya Teoriya Differencial′nyh Uravneniĭ, OGIZ, Moscow-Leningrad, 1947 (Russian). MR 0029483

J. v. Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. 36 (1934), no. 3, 445–492. MR 1501752, DOI 10.1090/S0002-9947-1934-1501752-3

Louis Padulo and Michael A. Arbib, System theory, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1974. A unified state-space approach to continuous and discrete systems. MR 0349260

J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 503–505. MR 348795, DOI 10.1090/S0002-9904-1974-13470-1

George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908

G. R. Sell, Book reviews, Bull. Amer. Math. Soc. 82 (1976), 198-207.

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1

Philippe Tondeur, Introduction to Lie groups and transformation groups, Lecture Notes in Mathematics, vol. 7, Springer-Verlag, Berlin-New York, 1965. MR 0193175

Ta-sun Wu, Distal minimal flow is coalescent, Proc. Amer. Math. Soc. 19 (1968), 429–432. MR 222877, DOI 10.1090/S0002-9939-1968-0222877-7

J. Auslander, Endomorphism of minimal sets, Duke Math. J. 30 (1963), 605-614. MR 27 #5245. MR 0155311




L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Ann. of Math. Studies, no. 53, Princeton Univ. Press, Princeton, N.J., 1963. MR 29 #4841.




L. Auslander, F. Hahn and L. Markus, Minimal flows on nilmanifolds, Flows on Homogeneous Spaces (Auslander, Green and Hahn, authors), Princeton Univ. Press, Princeton, N.J., 1963 (above).




N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Grundlehren Math. Wiss., Band 161, Springer-Verlag, Berlin and New York, 1970. MR 44 #7077. MR 324143




G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.




G. Birkhoff, Quelques théorèmes sur le mouvement des systemes dynamiques, Bull. Soc. Math. France 40 (1912), 1-19.




G. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., Vol. 9, Amer. Math. Soc., Providence, R.I., 1927. MR 209095




G. Birkhoff, What is the ergodic theorem?, Amer. Math. Monthly 49 (1942), 222-226. MR 6619




H. Bohr, Almost periodic functions, Chelsea, New York, 1947. MR 8, 512. MR 20163




G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 413144




R. W. Brockett, Lie algebras and Lie groups in control theory, Harvard Univ. (preprint).




M. L. Cartwright, From non-linear oscillations to topological dynamics, J. London Math. Soc. 39 (1964), 193-201. MR 28 #4197. MR 160988




E. J. Dubuc, Kan extensions in enriched category theory, Lecture Notes in Math., vol. 145, Springer-Verlag, Berlin and New York, 1970. MR 43 #6280. MR 280560




R. Ellis, Distal transformation groups, Pacific J. Math. 8 (1958), 401-405. MR 21 #96. MR 101283




R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc. 94 (1960), 272-281. MR 23 #A961. MR 123636




R. Ellis, Universal minimal sets, Proc. Amer. Math. Soc. 11 (1960), 540-543. MR 22 #8491. MR 117716




R. Ellis, The construction of minimal discrete flows, Amer. J. Math. 87 (1965), 564-574. MR 32 #3053. MR 185589




R. Ellis, Lectures on topological dynamics, Benjamin, New York, 1969. MR 42 #2463. MR 267561




R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94 (1960), 258-271. MR 23 #A960. MR 123635




A. M. Fink, Almost periodic differential equations, Lecture Notes in Math., vol. 377, Springer-Verlag, Berlin and New York, 1974. MR 460799




E. E. Floyd, A nonhomogeneous minimal set, Bull. Amer. Math. Soc. 55 (1949), 957-960. MR 11, 453. MR 33535




W. H. Gottschalk, Minimal sets: an introduction to topological dynamics, Bull. Amer. Math. Soc. 64 (1958), 336-351. MR 20 #6484. MR 100048




W. H. Gottschalk, Substitution minimal sets, Trans. Amer. Math. Soc. 109 (1963), 467-491. MR 32 #8325. MR 190915




W. H. Gottschalk, A survey of minimal sets, Ann. Inst. Fourier (Grenoble) 14 (1964), fasc. 1, 53-60. MR 29 #4044. MR 166771




W. H. Gottschalk, Bibliography for topological dynamics, 3rd ed., Wesleyan University, Middletown, Conn., 1967. MR 36 #7126. MR 224079




W. H. Gottschalk, Some general dynamic notions, Recent Advances in Topological Dynamics, Lecture Notes in Math., vol. 318, Springer-Verlag, Berlin and New York, 1973, pp. 120-125. MR 407821




W. H. Gottschalk and G. A. Hedlund, Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R.I., 1955. MR 17, 650. MR 74810




O. Hájek, Dynamical systems in the plane, Academic Press, New York, 1968. MR 39 #1767. MR 240418




J. L. Kelley, General topology, Van Nostrand, Princeton, N.J., 1955. MR 16, 1136. MR 70144




H. B. Keynes, On the proximal relation being closed, Proc. Amer. Math. Soc. 18 (1967), 518-522. MR 35 #3645. MR 212780




S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math., no. 5, Springer-Verlag, Berlin and New York, 1971. MR 50 #7275. MR 354798




E. G. Manes, A characterization of dual spaces, J. Functional Analysis 17 (1974), 292-295. MR 383048




E. G. Manes, Algebraic theories, Graduate Texts in Math., no. 26, Springer-Verlag, Berlin and New York, 1976. MR 419557




L. Markus, Lectures in differentiable dynamics, C.B.M.S. Regional Conf. Ser. in Math., no. 3, Amer. Math. Soc., Providence, R.I., 1971. MR 46 #8262. MR 309152




B. Mitchell, Theory of categories, Academic Press, New York, 1965. MR 34 #2647. MR 202787




D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York, 1955. MR 17, 383. MR 73104




V. V. Nemyckiĭ and V. V. Stepanov, Qualitative theory of differential equations, OGIZ, Moscow, 1947; English transl., Princeton Univ. Press, Princeton, N.J., 1960. MR 10, 612; 22 #12258. MR 29483




J. von Neumann, Almost periodic functions in a group. I, Trans. Amer. Math. Soc. 36 (1934), 445-492. MR 1501752




L. Padulo and M. A. Arbib, System theory, Saunders, Philadelphia, 1974. MR 50 # 1754. MR 349260




J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 503-505. MR 50 #1290. MR 348795




G. R. Sell, Topological dynamics and ordinary differential equations, Ann. of Math. Studies, no. 33, Van Nostrand, Reinhold, Princeton, N.J., 1971. MR 442908




G. R. Sell, Book reviews, Bull. Amer. Math. Soc. 82 (1976), 198-207.




S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 37 #3598; 39, p. 1593. MR 228014




P. Tondeur, Introduction to Lie groups and transformation groups, Lecture Notes in Math., vol. 7, Springer-Verlag, Berlin and New York, 1965. MR 33 # 1396. MR 193175




T.-S. Wu, Distal minimal flow is coalescent, Proc. Amer. Math. Soc. 19 (1968), 429-432. MR 36 #5927. MR 222877