## Institute for Mathematical Sciences

## Preprint ims91-17

** M. Lyubich and A. Volberg**
* A Comparison of Harmonic and Balanced Measures on Cantor Repellors *

Abstract: Let *J* be a Cantor repellor of a conformal map *f*. Provided *f* is a polynomial-like or *R*-symmetric, we prove that harmonic measure on *J* is equivalent to the measure of maximal entropy if and only if *f* is conformally equivalent to a polynomial. We also show that this is not true for general Cantor repellors: there is a non-polynomial algebraic function generating a Cantor repellor on which above two measures coincide.

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