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In this thesis we study the limiting dynamics of certain unbounded sequences in the moduli space of quadratic rational maps on . Given and c in the Mandelbrot set, let denote the unique quadratic rational map (up to conjugacy) with an attracting fixed point of multiplier λ that is hybrid equivalent to . Petersen proved that if c is in the p/q-limb of the Mandelbrot set, then tends to infinity in as λ tends to radially. Epstein showed that after normalizing and passing to a subsequence, the q-th iterate converges to a degree two rational map of the form .

Our main goal is to determine how the limiting map GT depends on c. We prove that in the real case, where , the answer depends on whether is renormalizable or not, i.e. whether or not. If is renormalizable then the limiting map GT is real hybrid equivalent to . For all non-renormalizable parameters the limit is G−2. We also show that as λ tends to -1 the Julia set of (suitably normalized) converges to the Julia set of the limiting map GT .

In preparation for these results, we prove that quite generally the quadratic limiting map of the q-th iterate is unique (it is independent of the choice of normalization). This result allows us to give an alternative, dynamical definition of DeMarco’s compactification of . We also determine the ideal limit points of the Perₙ(0) curves in .