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Growth of the zeta function for a quadratic map and the dimension of the Julia set


We show that the zeta function for the dynamics generated by the map z --> z(2)+c, c < -2, can be estimated in terms of the dimension of the corresponding Julia set. That implies a geometric upper bound on the number of its zeros, which are interpreted as resonances for this dynamical systems. The method of proof of the upper bound is used to construct a code for counting the number of zeros of the zeta function. The numerical results support the conjecture that the upper bound in terms of the dimension of the Julia set is optimal.

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