A Comparison of Harmonic and Balanced Measures on Cantor Repellors

Let J be a Cantor repellor of a conformal map f. Provided f is 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 nonpolynomial algebraic function generating a Cantor repellor on which two measures coincide.