On Polynomials Sharing Preimages of Compact Sets, and Related Questions

Abstract.

In this paper we give a solution of the following problem: under what conditions on infinite compact sets $K_1, K_2 \subset {\mathbb{C}}$ and polynomials f ₁, f ₂ do the preimages f ⁻¹₁ {K ₁} and f ⁻¹₂ {K ₂} coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets.