Let $X$ be a complete Kobayashi hyperbolic complex manifold. It is well-known that the automorphism group of $X$ is a real Lie group where the topology on the automorphim group is the compact-open topology. Suppose $f:X \to X$ is an automorphism with a fixed point $a$ and that some subsequence of the iterates of $f$ converges to the identity map in the compact-open topology. Then is it true that this subsequence converges to the identity uniformly on $X$ where the metric on $X$ is the Kobayashi metric?
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