Article PDF
References
Apostol, Tom M.: Mathematical analysis. A modern approach to advanced calculus. Reading, Mass. and London: Addison-Wesley Publishing Company, Inc. 1960.
Billingsley, Patrick: Ergodic theory and information. New York-London-Sidney: John Wiley & Sons, Inc. 1965.
Bissinger, B.H.: A generalization of continued fractions. Bull. Amer. math. Soc. 50, 868–876 (1944).
Chung, K.L.: A note on the ergodic theorem of information theory. Ann. math. Statistics 32, 612–614 (1961).
Dunford, N., Miller, D.S.: On the ergodic theorem. Trans. Amer. math. Soc. 60, 538–549 (1946).
Everett, G.I.: Representations for real numbers. Bull. Amer. math. Soc. 52, 861–869 (1946).
Ibragimov, I.A.: Some limit theorems for stochastic processes stationary in the strict sense. Doklady Akad. Nauk. SSSR u. Sev. 125, 711–714 (1959): English transl. Select. Transl. math. Statist. and Probab. 4, 99–103 (1963).
—: Asymptotic distribution of the values of certain sums. Vestnik Leningrad. Univ. 15, 7, 81–115 (1960): English transl. Select. Transl. math. Statist. and Probab. 5, 33–48 (1965).
—: Some limit theorems for stationary processes. Theor. Probab. Appl. 7, 4, 349–382 (1962).
Jacobi, C.G.J.: Allgemeine Theorie der kettenbruchÄhnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird. J. reine angew. Math. 69, 29–64 (1868).
Khinchin, A. Ya: Continued fractions. Chicago and London: The University of Chicago Press 1964.
Kinney, J., Pitcher, T.: The dimension of some sets defined in terms of f-expansions. Z. Wahrscheinlichkeitstheorie verw. Geb. 4, 293–315 (1966).
Knopp, K.: Mengen theoretische Behandlung einiger Probleme der diophantischen Approximationen und der transfiniten Wahrscheinlichkeiten. Math. Ann. 95, 409–426 (1926).
Kuzmin, R.O.: Sur un probleme de Gauss. Atti del Congresso Internazionale del Matematici Bologna 6, 83–89 (1928).
Parry, W.: On Rohlin's formula for entropy. Acta math. Acad. Sci. Hungar. 15, 107–113 (1964).
Parry, William: Entropy and generators in ergodic theory. New York and Amsterdam: Benjamin, Inc. 1969.
Perron, Oskar: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann. 64, 1–76 (1907).
Renyi, A.: Representations for real numbers and their ergodic properties. Acta math. Acad. Sci. Hungar. 8, 477–493 (1957).
Reznik, M.H.: The law of the iterated logarithm for some classes of stationary processes (in Russian). Teor. Verojatn. Primen. 8, 4, 642–656 (1968).
Rohlin, V.A.: Exact endomorphisms of a Lebesgue space. Izvestya Akad. Nauk. SSSR, Ser. mat. 24, 499–530 (1960): English Transl. Amer. math. Soc. Translat. II Ser. 39, 1–36 (1964).
Ryll-Nardzewski, C.: On the ergodic theorems (II). Ergodic theory of continued fractions. Studia math. 12, 74–79 (1951).
Saks, S.: Theory of the integral. New York: Hafner Publishing Company 1937.
Schweiger, F.: Geometrische und elementar metrische SÄtze über den Jacobischen Algorithmus. österr. Akad. Wiss. Wien, math.-naturw. Kl., S.-Ber. Abt. II, 173, 59–92 (1964).
—: Metrische SÄtze über den Jacobischen Algorithmus. Monatsh. Math. 69, 243–255 (1965).
—: Ergodische Theorie des Jacobischen Algorithmus. Acta arithmetica 11, 451–460 (1966).
—: Existenz eines invarianten Masses beim Jacobischen Algorithmus. Acta arithmetica 12, 263–268 (1967).
—: Mischungseigenschaften und Entropie beim Jacobischen Algorithmus. J. reine angew. Math. 229, 50–56 (1968).
—: Induzierte Masse und Jacobischen Algorithmus. Acta arithmetica 13, 419–422 (1968).
—: Ein Kuzminscher Satz über den Jacobischen Algorithmus. J. reine angew. Math. 232, 35–40 (1968).
—: Metrische Theorie einer Klasse zahlentheoretischer Transformationen. Acta arithmetica 15, 1–18 (1968).
Vinh-Hien, Tran.: The central limit theorem for stationary processes generated by number theoretic endomorphisms (in Russian). Vestnik Moskov Univ. Ser. I. Mat. Meh. 5, 1 28–34 (1963).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Waterman, M.S. Some ergodic properties of multi-dimensional f-expansions. Z. Wahrscheinlichkeitstheorie verw Gebiete 16, 77–103 (1970). https://doi.org/10.1007/BF00535691
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00535691