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.

Details: 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} $.

I haven't read a good tutorial yet, so I hope I didn't get the notations or the essence wrong.

Thanks

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

I'm looking for a coincise 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.

Details: 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} $.

I haven't read a good tutorial yet, so I hope I didn't get the notations or the essence wrong.

Thanks

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.

Details: 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} $.

I haven't read a good tutorial yet, so I hope I didn't get the notations or the essence wrong.

Thanks