Fractional iteration of two-dimensional continuous mappings

The existence of continuous fractional iterates (iterative roots) for a class of two-dimensional homeomorphisms is investigated. The theory of iterative roots for one-dimensional monotone mappings is first presented. Subsequently, an analysis is conducted on two-dimensional homeomorphisms satisfying specific conditions, where the linear part is a hyperbolic invertible matrix and the nonlinear perturbation is a Lipschitz continuous bounded function. The results indicate that when one eigenvalue of the matrix lies between 0 and 1 and the other is greater than 1, the mapping possesses homeomorphic iterative roots of any positive integer order. When the two eigenvalues satisfy other specific combinations, the mapping also admits homeomorphic iterative roots for all positive odd integer orders.