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Fat baker's transformations

Published online by Cambridge University Press:  19 September 2008

J. C. Alexander  and
J. A. Yorke
J. C. Alexander
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
J. A. Yorke
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA
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Abstract

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We investigate a variant of the baker's transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 1 , March 1984 , pp. 1 - 23
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Balatoni, J. & Rényi, A.. Remarks on entropy, (in Hungarian with English and Russian summaries). Publ. of Math. Inst. Hungarian Acad. Aci. (1) (1956), 9–40.Google Scholar
(See also Rényi, A., Dimension, entropy and information, Transactions of the Second Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, (1957), 545–556;Google Scholar
see also Rényi, A., On the dimension and entropy of probability distributions, Acta Math. Hung. 10 (1959), 193–215.)Google Scholar
[2]Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J-M.. Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them, Part 1: Theory. Meccanica 15 (1980), 9–20;Google Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J-M.. Lyapunov characteristic exponents for smooth dynamical systems; a method for computing all of them, Part 2: Numerical application, Meccanica, 21–29.Google Scholar
[3]Billingsley, P.. Probability and Measure. J. Wiley, 1979.Google Scholar
[4]Bowen, R.. A model for Couette flow data. In Turbulence Seminar, Springer-Verlag Lecture Notes in Mathematics #615, (1977), 117–134.Google Scholar
[5]Bowen, R. & Ruelle, D.. The ergodic theory of axiom-A flows. Invent. Math. 29 (1975), 181–202.Google Scholar
[6]Boyarsky, A.. On the significance of absolutely continuous invariant measures. Preprint: Concordia Univ., 1982.Google Scholar
[7]Erdös, P.. On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 (1939), 974–976.Google Scholar
[8]Erdös, P.. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math. 62 (1940), 180–186.Google Scholar
[9]Farmer, D.. Chaotic attractors of an infinite dimensional dynamical system. Phys. D, 4 (1982), to appear.Google Scholar
[10]Frederickson, P., Kaplan, J. L., Yorke, E. D. & Yorke, J. A.. The Lyapunov dimension of strange attractors. J. Differential Equations, to appear, 1982.Google Scholar
[11]Garsia, A. M.. Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409–432.Google Scholar
[12]Garsia, A. M.. Entropy and singularity of infinite convolutions. Pacific J. Math. 13 (1963), 1159–1169.Google Scholar
[13]Hanson, J. E., Ott, E. & Russell, D. A.. Dimensionality and Lyapunov numbers of strange attractors. Phys. Rev. Lett. 45 (1980), 1175–1178.Google Scholar
[14]Jessen, B. & Wintner, A.. Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc. 38 (1938), 48–88.Google Scholar
[15]Kahane, J. P.. Sur la distribution de certaines series aleatoires. Bull. Soc. Math. France 25 (1971), 119–122.Google Scholar
[16]Kaplan, J. L. & Yorke, J. A.. Chaotic behaviour of multidimensional difference equations. In Functional Differential Equations and Approximation of Fixed Points, (Peitgen, H-O. and Walther, H-O., eds.). Springer-Verlag Lecture Notes in Mathematics #730 (1979), 223–237.Google Scholar
[17]Kaplan, J. L., Mallet-Paret, J. & Yorke, J. A.. The Lyapunov dimension of a nowhere differentiable torus. Ergod. Th. & Dynam. Sys. To appear.Google Scholar
[18]Kawata, T.. Fourier Analysis in Probability Theory. Academic Press, 1972.Google Scholar
[19]Keller, G.. Ergodicité et mesures invariantes pour les transformations dilantes par morceaux d' région bornée du plan. C.R. Acad. Sc. Paris, 289 (1979), 625–627.Google Scholar
[20]Ledrappier, F.. Some relations between dimension and Lyapunov exponents. Comm. Math. Phys. 81 (1981), 229–238.Google Scholar
[21]Oseledec, V. I.. A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trudy Moskov. Math. Obsch. 19 (1968), 179–210.Google Scholar
(English translation: Trans. Mosc. Math. Soc. 19 (1968), 197–231.)Google Scholar
[22]Salem, R.. A remarkable class of algebraic integers, proof of a conjecture of Vijayaraghavan. Duke Math. J. 11 (1944), 103–108.Google Scholar
[23]Salem, R.. Algebraic Numbers and Fourier Analysis. Heath., 1963.Google Scholar
[24]Siege, C. L.). Algebraic integers whose conjugates lie in the unit circle. Duke Math. J. 11 (1944), 597–602.Google Scholar
[25]Walters, P.. Ergodic Theory. Springer-Verlag Lecture Notes in Mathematics # 458, Springer-Verlag: Berlin-Heidelberg-New York, (1975).Google Scholar
[26]Wintner, A.. On convergent Poisson convolutions. Amer. J. Math. 57 (1935), 827–838.Google Scholar
[27]Young, L-S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys. 2 (1982), 109–124.Google Scholar