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I'm looking for a concise introductory on the subject of beta transformations on the circle. I've found things that are related to its applications in the computational field or in number theory. A good intro for me will include basic definitions, main results about the transformations themselves and references.

Basic Notion: If $X=[0,1]$, then for all $\beta >0$ the associated $\beta$ transformation is $T_{\beta} x = \beta x \, {\rm mod} (1)$, where $x\in X$. This can be easily embeded in the unit circle $\mathbb{T}$ by $T_{\beta} e^{2\pi i x} = e^{2\pi i \beta x} $.

The above formulation is not exact, as was implied in the comments. Probably this is why I'm in need of a tutorial.

Thanks

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    $\begingroup$ Is it well defined on the circle? If I replace $x$ by $x+k$ for an integer $k$, I should have the same left hand side. But if $\beta$ is not an integer, I seem to have a different right hand side. $\endgroup$
    – Ben McKay
    Commented Mar 30, 2017 at 13:59
  • $\begingroup$ @BenMcKay from the references given, I see your point. However, I don't think I'm yet in the position to give the right formulation in the OP. $\endgroup$
    – Amir Sagiv
    Commented Mar 30, 2017 at 14:53

3 Answers 3

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Everything I know about beta-transformations I learned from Charlene Kalle, whose PhD thesis (Utrecht University, 2009) can be downloaded here: http://www.math.leidenuniv.nl/~kallecccj/proefschrift.pdf. It is interesting, well written and even funny at times, but of course being a thesis it focuses on proving new results rather than providing an introduction to the subject. However, the first chapter might be a good place to start. It does include basic definitions and main results about the transformations and the thesis as a whole includes a lot of references.

Another book that might fit the bill is reference [DK02a] in the thesis:

K. Dajani and C. Kraaikamp. Ergodic theory of numbers, volume 29 of Carus Mathematical Monographs. Mathematical Association of America, Washington, DC, 2002.

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    $\begingroup$ I think a better place to start is where the notion is firstly proposed, say Renyi's paper. $\endgroup$
    – Henry.L
    Commented Mar 30, 2017 at 15:29
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I think there are several papers that can give you a good introduction. The seminal papers of Parry and Rényi have been already mentioned as well as Dajani-Kraikaamp's book. I think that Nikita Sidorov's paper Arithmethic Dynamics is very useful as well as his lecture notes. Also Dajani&Kraaikamp's paper From Greedy to Lazy expansions and their driving dynamics is relevant as well as Komornik's paper Expansions in noninteger bases.

Now, Simon Baker's and Derong Kong's papers contain very well written brief introductions to the topic.

EDIT I found these notes today. http://wisdyn.math.uni-bremen.de/betabremen.pdf http://wisdyn.math.uni-bremen.de/Lingmin.pdf

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The reference pointed out by Vincent are both thesis, I think a more appropriate introductory level material is Brown&Yin. And it is the only modern paper that I touched tangential to beta expansion.

[Brown&Yin]Brown, Gavin, and Qinghe Yin. "$\ beta $-transformation, natural extension and invariant measure." Ergodic Theory and Dynamical Systems 20.05 (2000): 1271-1285.

Another source is Renyi's original paper where $\beta$-transformation is first invented.

Rényi, Alfréd. "Representations for real numbers and their ergodic properties." Acta Mathematica Hungarica 8.3-4 (1957): 477-493.

and a follow-up paper,

Parry, William. "On the β-expansions of real numbers." Acta Mathematica Hungarica 11.3-4 (1960): 401-416.

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