This thesis consists of five papers, each of which investigates the dynamics of multivalued maps on the Riemann sphere or in the complex plane. The problems studied stem from, and are special cases of, the Pólya–Schur problem.

In Papers I and III, we initiate the study of continuous Hutchinson invariance and characterize the associated invariant sets both topologically and geometrically. In Paper I, we establish necessary and sufficient conditions for the existence of a unique invariant set that is minimal under inclusion, and determine when this set is non-trivial and when it is compact. Paper III focuses on the boundary of this set, providing bounds on its complexity in a specific sense. Moreover, we classify the different types of boundary points.

Papers II and IV examine the dynamics of holomorphic correspondences. In Paper II, we construct explicit differential operators and associated holomorphic correspondences such that, for the differential operator, there exists a unique Hutchinson-invariant set in high degrees that is minimal under inclusion, and we study the equidistribution of the associated holomorphic correspondences. In Paper IV, we explore conformal measures of a class of (anti)holomorphic correspondences, prove their existence, and derive bounds on the Hausdorff dimension of the limit sets.

Paper V focuses on a question more closely related to the Pólya--Schur theory. For a given differential operator T and a degree n, a set S ⊂ ℂ is said to be T_n-invariant if T maps any polynomial of degree n with all zeros in S to a polynomial with all zeros in S, or to the zero polynomial. We find conditions on T that guarantee the existence of a unique T_n-invariant set that is minimal under inclusion and establish some of its properties.