Abstract
A trajectory of a system which is evolving with time may be said to be ‘recurrent’ if it keeps returning to any neighbourhood, however small, of its initial point, and ‘dense’ if it passes arbitrarily near to every point. It may be said to be ‘uniformly distributed’ if the proportion of time it spends in any region tends asymptotically to the ratio of the volume of that region to the volume of the whole space. In the present chapter these notions will be made precise and some fundamental properties derived. The subject of dynamical systems has its roots in mechanics, but we will be particularly concerned with its applications in number theory.
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Selected References
V.I. Arnold and A. Avez, Ergodic problems of classical mechanics, Benjamin, New York, 1968.
K.I. Babenko, On a problem of Gauss, Soviet Math. Dokl. 19 (1978), 136–140.
J. Beck, Probabilistic diophantine approximation, I. Kronecker sequences, Ann. of Math. 140 (1994), 451–502.
J. Beck and W.W.L. Chen, Irregularities of distribution, Cambridge University Press, 1987.
V. Bergelson and A. Leibman, Set polynomials and polynomial extension of the Hales–Jewett theorem, Ann. of Math. 150 (1999), 33–75.
P. Billingsley, Probability and measure, 3rd ed., Wiley, New York, 1995.
P. Billingsley, Ergodic theory and information, reprinted, Krieger, Huntington, N.Y., 1978.
G. Brown and W. Moran, Schmidt's conjecture on normality for commuting matrices, Invent. Math. 111 (1993), 449–463.
H.E. Buchanan and H.T. Hildebrandt, Note on the convergence of a sequence of functions of a certain type, Ann. of Math. 9 (1908), 123–126.
K. Chandrasekharan, Exponential sums in the development of number theory, Proc. Steklov Inst. Math. 132 (1973), 3–24.
Y.-G. Chen, The best quantitative Kronecker's theorem, J. London Math. Soc. (2) 61 (2000), 691–705.
I.P. Cornfeld, S.V. Fomin and Ya. G. Sinai, Ergodic theory, Springer-Verlag, New York, 1982.
M. Drmota and R.F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics 1651, Springer, Berlin, 1997.
I. Dupain and V.T. Sós, On the discrepancy of (nα) sequences, Topics in classical number theory (ed. G. Halász), Vol. I, pp. 355–387, North-Holland, Amsterdam, 1984.
H. Dym and H.P. McKean, Fourier series and integrals, Academic Press, Orlando, FL, 1972.
P. and T. Ehrenfest, The conceptual foundations of the statistical approach in mechanics, English translation by M.J. Moravcsik, Cornell University Press, Ithaca, 1959. [German original, 1912]
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981.
H. Furstenberg and Y. Katznelson, A density version of the Hales–Jewett theorem, J. Analyse Math. 57 (1991), 64–119.
W.T. Gowers, A new proof of Szemeredi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.
R.L. Graham, B.L. Rothschild and J.H. Spencer, Ramsey theory, 2nd ed., Wiley, New York, 1990.
S.W. Graham and G. Kolesnik, Van der Corput’s method of exponential sums, London Math. Soc. Lecture Notes 126, Cambridge University Press, 1991.
P.R. Halmos, Measure theory, 2nd printing, Springer-Verlag, New York, 1974.
D.M. Hardcastle and K. Khanin, Continued fractions and the d-dimensional Gauss transformation, Comm. Math. Phys. 215 (2001), 487–515.
B. Jessen and H. Tornehave, Mean motion and zeros of almost periodic functions, Acta Math. 77 (1945), 137–279.
A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, 1995.
Y. Katznelson and B. Weiss, A simple proof of some ergodic theorems, Israel J. Math. 42 (1982), 291–296.
J.H.B. Kemperman, Distributions modulo 1 of slowly changing sequences, Nieuw Arch. Wisk. (3) 21 (1973), 138–163.
J.F.C. Kingman, Subadditive processes, Ecole d’Eté de Probabilités de Saint-Flour V-1975 (ed. A. Badrikian), pp. 167–223, Lecture Notes in Mathematics 539, Springer-Verlag, 1976.
U. Krengel, Ergodic theorems, de Gruyter, Berlin, 1985.
L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974.
J.C. Lagarias, The 3x + 1 problem and its generalizations, Amer. Math. Monthly 92 (1985), 3–23.
M. Loève, Probability theory, 4th ed. in 2 vols., Springer-Verlag, New York, 1978.
D.H. Mayer, On the thermodynamic formalism for the Gauss map, Comm. Math. Phys. 130 (1990), 311–333.
G. Mills, A quintessential proof of van der Waerden’s theorem on arithmetic progressions, Discrete Math. 47 (1983), 117–120.
H.L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics 84, American Mathematical Society, Providence, R.I., 1994.
H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), 957–1041.
H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS–NSF Regional Conference Series in Applied Mathematics 63, SIAM, Philadelphia, 1992.
H. Niederreiter and W. Philipp, Berry–Esseen bounds and a theorem of Erdös and Turán on uniform distribution mod 1, Duke Math. J. 40 (1973), 633–649.
K. Petersen, Ergodic theory, Cambridge University Press, 1983.
W. Philipp and O.P. Stackelberg, Zwei Grenzwertsätze für Kettenbrüche, Math. Ann. 181 (1969), 152–156.
H. Poincaré, Sur la théorie cinétique des gaz, Oeuvres, t. X, pp. 246–263, Gauthier-Villars, Paris, 1954.
F. Riesz and B. Sz.-Nagy, Functional analysis, English transl. by L.F. Boron, Ungar, New York, 1955.
A. Rockett and P. Szusz, Continued fractions, World Scientific, Singapore, 1992.
W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, New York, 1976.
D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58.
S. Saks, Theory of the integral, 2nd revised ed., English transl. by L.C. Young, reprinted, Dover, New York, 1964.
S. Shelah, Primitive recursion bounds for van der Waerden numbers, J. Amer. Math. Soc. 1 (1988), 683–697.
J.M. Steele, Kingman’s subadditive ergodic theorem, Ann. Inst. H. Poincaré Sect. B 25 (1989), 93–98.
M.H. Stone, A generalized Weierstrass approximation theorem, Studies in modern analysis (ed. R.C. Buck), pp. 30–87, Mathematical Association of America, 1962.
B.L. van der Waerden, How the proof of Baudet’s conjecture was found, Studies in Pure Mathematics (ed. L. Mirsky), pp. 251–260, Academic Press, London, 1971.
P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982.
H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), 313–352. [Reprinted in Selecta Hermann Weyl, pp. 111–147, Birkhäuser, Basel, 1956 and in Hermann Weyl, Gesammelte Abhandlungen (ed. K. Chandrasekharan), Band I, pp. 563–599, Springer-Verlag, Berlin, 1968]
E. Wirsing, On the theorem of Gauss–Kusmin–Lévy and a Frobenius type theorem for function spaces, Acta Arith. 24 (1974), 507–528.
R.J. Zimmer, Ergodic theory and semi-simple groups, Birkhäuser, Boston, 1984.
Additional Reference
B. Kra, The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view, Bull. Amer. Math. Soc. (N.S.) 43 (2006), 3–23.
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Coppel, W.A. (2009). Uniform Distribution and Ergodic Theory. In: Number Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89486-7_11
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