We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order ell of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to 0 under the dynamics of the tower for corresponding ell. That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when ell tends to infinity. We also prove the convergence of the drifts to a finite limit which can be expressed purely in terms of the limiting tower which corresponds to a Feigenbaum map with a flat critical point
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