Complex rotation numbers

Take a cylinder of height $h$ and glue its boundary circles via the rotation by angle $a$. We get a torus with inherited complex structure. The modulus of this elliptic curve is $a+ih$. Now, let us glue the boundary circles of the cylinder via some analytic circle diffeomorphism $f$ instead of the rotation. Once again, we get an elliptic curve. What happens to its modulus as $h$ tends to $0$? (this problem and this construction were suggested by V. I. Arnold, 1978). The answer depends on the dynamical properties of the circle diffeomorphism $f$ and is related to its rotation number. This answer leads to a new interesting set (“bubbles”), analogue of Arnold tongues. The talk is devoted to a description of the bubbles.