## Institute for Mathematical Sciences

## Preprint ims90-8

** J. J. P. Veerman and F. M. Tangerman **
* Scalings in Circle Maps (I)*

Abstract: Let f be a flat spot circle map with irrational rotation number. Located at the edges of the flat spot are non-flat critical points (S: x -> Ax^{v} ,v≥1). First, we define scalings associated with the closest returns of the orbit of the critical point. Under the assumption that these scalings go to zero, we prove that the derivative of long iterates of the critical value can be expressed in the scalings. The asymptotic behavior of the derivatives and the scalings can then be calculated. We concentrate on the cases for which one can prove the above assumption. In particular, let one of the singularities be linear. These maps arise for example as the lower bound of the non-decreasing truncations of non-invertible bimodal circle maps. It follows that the derivatives grow at a sub-exponential rate.

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