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In a previous work, we constructed a positive-definite total energyfunctional for the axially symmetric linear perturbative theory of Kerr blackhole spacetimes. That work is based on the dimensional reduction of dynamicalaxisymmetric spacetimes into 2+1 Einstein-wave map system. In the constructionof the positive-definite energy, various dynamical terms, at the boundary ofthe orbit space, critically occur.  In this work, after setting up the initial value problem in harmoniccoordinates, we prove that the positive energy for the axially symmetric linearperturbative theory of Kerr black holes is strictly conserved in time, byestablishing that all the boundary terms dynamically vanish for all times. Thisresult implies a form of dynamical linear stability of Kerr black holes

Axially Symmetric Perturbations of Kerr Black Holes II: Boundary Behaviour of the Dynamics in the Orbit Space