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In order to study the global behavior of difference equations, it is necessary to understand the local behavior in a neighborhood of a equilibrium point of the difference equation. This thesis focuses on two aspects of the local behavior of planar difference equations: the asymptotic behavior of a solution converging to a hyperbolic fixed point, and the local qualitative behavior of a non isolated fixed point whose jacobian matrix has a particular structure. Manuscript 2 describes how closely a convergent solution {xn} of (real or complex) difference equations xn+1 = J xn + fn(xn) can be approximated by its linearization zn+1 = J zn in a neighborhood of a fixed point; where xn is a m- vector, J is a constant m×m matrix and fn(y) is a vector valued function which is continuous in y for fixed n, and where fn(y) is small in a sense. Manuscript 3 describes completely the local qualitative behavior of a real planar map in a neighborhood of a non-isolated fixed point whose jacobian matrix is similar to ( 1 1 0 1 ), also called a non-isolated 1-1 resonant fixed point. Theorem 3 gives conditions for four non-conjugate dynamical scenarios to occur.

PLANAR DIFFERENCE EQUATIONS: ASYMPTOTIC BEHAVIOR OF SOLUTIONS AND 1-1 RESONANT POINTS