Projections in Rotation Algebras and Theta Functions

For each $\alpha \in (0,1)$, $A_\alpha$ denotes the universal $C^*$-algebra generated by two unitaries $u$ and $v$, which satisfy the commutation relation $uv=\exp (2\pi i\alpha)vu$. We consider the order four automorphism $\sigma$ of $A_\alpha$ defined by $\sigma (u)=v$, $\sigma (v)=u^{-1}$ and describe a method for constructing projections in the fixed point algebra $A_\alpha^\sigma$, using Rieffel's imprimitivity bimodules and Jacobi's theta functions. In the case $\alpha =q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae for such projections and find a lower bound for the norm of the Harper operator $u+u^* +v+v^*$.Comment: Fixes some numerical issues in the published versio

On certain statistical properties of continued fractions with even and with odd partial quotients

We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's \alpha-expansions.Comment: Final versio