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Random β-transformations

   School of Mathematics

  Dr Tony Samuel  Applications accepted all year round  Competition Funded PhD Project (Students Worldwide)

About the Project

One of the most classical examples of a dynamical system stem from β-transformations. Namely, self-maps on [0, 1] of the form x → {βx}, where β is a real number with |β| > 1 and where {βx} denotes the fractional part of βx. They are the simplest family of expanding interval maps and yet give rise to a variety of intricate dynamical behaviours. When β is not an integer, these transformations lead to a beautiful representation of real numbers, i.e. non-integer base expansions, which have their origins in the works of Alexander Gel'fond [1], William Parry [4] and Alfréd Rényi [5]. These β-expansions possess very different properties to their integer counterparts (e.g. binary or decimal), and continued fraction expansions. By exploiting these discrepancies Yutaka Jitsumatsu et al. [3, 4] has developed a new random number generator which was verified by the National Institute of Standards and Technology (NIST) statistical test suite. This is where this project will kick off, namely to explore the results of Yutaka Jitsumatsu et al. and to tackle some of the open questions posed in [3, 4] relating to random dynamical systems, in particular system driven by random β-transformations. This project is planned to be supervised by Dr C. Kalle (Universiteit Leiden, The Netherlands) and Dr T. Samuel (University of Birmingham, UK), and applicants will have the option to be based either at the University of Birmingham or Universiteit Leiden with long term visits to the partner institution.

Entry requirements

We are looking for a creative and enthusiastic graduate with:

  • a strong mathematics profile demonstrated by, for instance, achieving a strong Masters degree (or equivalent) in mathematics or a closely related discipline,
  • a solid background in analysis, ergodic theory, measure theory and/or numeration systems, and
  • excellent communication skills (English).

Application process and enquiries

For applicants wanting to be based at the University of Birmingham, the application procedure and the deadlines for scholarships are available at https://www.birmingham.ac.uk/schools/mathematics/phd/phd-programme.aspx. For applicants wanting to be based at Universiteit Leiden, please contact Dr C. Kalle.

Informal enquiries should be addressed to Dr C. Kalle or Dr T. Samuel – contact details are available on their respective homepages.

Funding Notes

For applicants wanting to be based at the University of Birmingham, a funded scholarship is available for candidates demonstrating exceptional capabilities. The scholarship typically covers a stipend at standard UKRI rates for 3-3.5 years, and in some cases the cost of tuition fees.


[1] A. O. Gel’fond. A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat. 23, 809–814 (1959).
[2] K. Itaya and Y. Jitsumatsu. Random Number Generation Using Outputs from Multiple Beta Encoders, IEICE Proceedings Series 48, A4L-C-6 (2016).
[3] Y. Jitsumatsu and K. Matsumura. A β-ary to binary conversion for random number generation using a β encoder, Nonlinear Theory and Its Applications, IEICE 7(1), 38-55 (2016).
[4] W. Parry. On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960).
[5] A. Réenyi. Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8, 477–493 (1957).

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