# Non literal tranducers and some problems of normality

Journal de théorie des nombres de Bordeaux (1993)

- Volume: 5, Issue: 2, page 303-321
- ISSN: 1246-7405

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topBlanchard, François. "Non literal tranducers and some problems of normality." Journal de théorie des nombres de Bordeaux 5.2 (1993): 303-321. <http://eudml.org/doc/93584>.

@article{Blanchard1993,

abstract = {A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base $p^k$ (resp. $p$) are also near normality to base $p$ (resp. $p^k$), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.},

author = {Blanchard, François},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {transducers; normality; real numbers; base changes; automata; Maxfield's theorem},

language = {eng},

number = {2},

pages = {303-321},

publisher = {Université Bordeaux I},

title = {Non literal tranducers and some problems of normality},

url = {http://eudml.org/doc/93584},

volume = {5},

year = {1993},

}

TY - JOUR

AU - Blanchard, François

TI - Non literal tranducers and some problems of normality

JO - Journal de théorie des nombres de Bordeaux

PY - 1993

PB - Université Bordeaux I

VL - 5

IS - 2

SP - 303

EP - 321

AB - A new proof of Maxfield’s theorem is given, using automata and results from Symbolic Dynamics. These techniques permit to prove that points that are near normality to base $p^k$ (resp. $p$) are also near normality to base $p$ (resp. $p^k$), and to study genericity preservation for non Lebesgue measures when going from one base to the other. Finally, similar results are proved to bases the golden mean and its square.

LA - eng

KW - transducers; normality; real numbers; base changes; automata; Maxfield's theorem

UR - http://eudml.org/doc/93584

ER -

## References

top- [A] L.M. Abramov, The entropy of a derived automorphism, Amer. Math. Soc. Transl.49 (1965), 162-166. Zbl0185.21804
- [BP] J. Berstel, D. Perrin, Theory of codes, London, Academic Press, 1985. Zbl0587.68066MR797069
- [Be] A. Bertrand-Mathis, Développements en base θ et répartition modulo 1 de la suite (xθn), Bull. Soc. math. Fr.114 (1986), 271-324. Zbl0628.58024
- [BIP] F. Blanchard, D. Perrin, Relèvements d'une mesure ergodique par un codage, Z. Wahrsheinlichkeist. v. Gebiete54 (1980), 303-311. Zbl0441.60004MR602513
- [BIDT] F. Blanchard, J.-M. Dumont, A. Thomas, Generic sequences, transducers and multiplication of normal numbers, Israel J. Math.80 (1992), 257-287. Zbl0776.11040MR1202572
- [BrL] A. Broglio, P. Liardet, Predictions with automata, Symbolic Dynamics and its Applications, AMS, providence, RI, P. Walters ed., Contemporary Math135, 1992. Zbl0777.11028MR1185084
- [C] J.W.S. Cassels, On a paper of Niven and Zuckerman, Pacific J. Math.3 (1953), 555-557. Zbl0047.04402MR51271
- [D] J.-M. Dumont, Private communication.
- [F1] C. Frougny, Representations of numbers and finite automata, Math. Systems Theory25 (1992), 37-60. Zbl0776.11005MR1139094
- [F2] C. Frougny, How to write integers in non-integer base, Lecture Notes in Computer Science (Proceedings of Latin '92) 583, 154-164. MR1253354
- [F3] C. Frougny, B. Solomyak, Finite Beta-expansions, Ergod. Th. & Dynam. Syst. 12 (1992), 713-723. Zbl0814.68065MR1200339
- [K] T. Kamae, Subsequences of normal sequences, Israel J. Math.16 (1973), 121-149. Zbl0272.28012MR338321
- [M] J.E. Maxfield, Normal k-tuples, Pacif. J. Math.3 (1953), 189-196. Zbl0050.27503MR53978
- [PS] K. Petersen, L. Shapiro, Induced flows, Trans. Amer. Math. Soc.177 (1973), 375-390. Zbl0229.54036MR322839