In 1978, V.Arnold suggested the following way to construct a complex torus for any analytic circle diffeomorphism $f$ and a complex number $\omega$. Namely, we glue the borders of the cylinder
$$\{0 \le \mathrm{Im}\ z \le \mathrm{Im}\ \omega,\ z \in \mathbb C/ \mathbb Z\},$$
by $f+\omega$. As $\omega \to \mathbb R$, the limit values of modulus of this complex torus form a so-called set "bubbles" in the upper half-plane. "Bubbles" reflect dynamical properties of circle diffeomorphisms $f+\omega, \omega \in \mathbb{R}$. Renormalizations of circle diffeomorphisms produce self-similarity patterns in the fractal-like picture of bubbles.