arXiv:math/0511444v4 [math.GT] 10 Aug 2006 DYNAMICAL TYPES OF ISOMETRIES OF THE HYPERBOLIC SPACE KRISHNENDU GONGOPADHYAY AND RAVI S. KULKARNI Abstract. In this paper after giving a finer classification, we give an algebraic characterization of the dynamical types of the isometries of the hyperbolic n-space H n . This has been done by using the linear representation of the isometry group of the hyperboloid model of H n . Using the representation of the isometries as 2 × 2 matrices over C and H, we give another algebraic characterization of the dynamical types of the isometries of H 3 and the orientation-preserving isometries of H 5 respectively. We also derive the parameter spaces of orientation-preserving isometries of H 5 with a fixed dynamical type. Contents 1. Introduction 2 2. The group O(V,Q) 3 2.1. Dimension of a maximal null-space 4 2.2. Tangent space to a point on the light-like cone 5 3. The isometries of H n and their dynamical types 5 3.1. Classification of isometries based on fixed-point 6 3.2. Dynamical types of isometries in H n 9 3.3. Conjugacy classes in PO(n, 1) 10 3.4. Algebraic characterization of the dynamical types 10 3.5. Upper-bound of rotation-angles 11 4. Effective procedures to determine the dynamical types 11 5. Algebraic characterization of dynamical types in low dimensions 13 5.1. Dynamical types in hyperbolic and Euclidean geometry 13 5.2. Hyperbolic and M¨ obius geometry 14 5.3. Dynamical types of isometries of H 3 14 5.4. Dynamical types of isometries of H 5 17 6. Parameter space of isometries of H 5 with a fixed dynamical type 24 References 28 Date : August 10, 2006. 1