| The theory of differential dynamical systems mainly concerns with the properties of the systems that change over time. Due to its wide and effective applications, it becomes the emerg-ing field of discipline that attracts increasing attention. In the theory of differential dynamical systems, the linearization of a mapping F means that there is a homeomorphism Φ satisfying the conjugacy equation Φ ο F = A ο Φ, where A is the linear part of F. Linearization allows us to understand dynamical properties of a nonlinear system by studying a linear one. It is one of the basic methods to study local qualitative properties of the systems.In the first part of the paper, we investigate the problem of linearization for planar map-pings. Although the famous Hartman Theorem proves that the diffeomorphism can be topo-logically conjugated to its linear part near the hyperbolic fixed point, in some applications, one further expects that the conjugacy Φ could be Holder continuous. This is the so-called problem of Holder linearization.For C1 and C1,1 mappings, the Holder exponents of the conjugacy have been estimated respectively. In this paper C1,α (α∈(0,1)) mappings, whose smoothness is be-tween C1 and C1,1 will be considered. By employing sequence of functions involving iterates of the given mappings to approximate the solutions of the conjugacy equation, we estimate Holder exponent of Φ to extend the previous results.In the second part, we are concerned with the existence of a class of iterative functional equations. We first relate solving the functional equations to the problem of invariant curves for mappings. Then, using known results on C1 linearization, we prove the existence of C1 solutions of the iterative functional equations under weaker conditions of coefficients. |
