Equivalence of Collet–Eckmann Conditions for Slowly Recurrent Rational Maps

The uploaded note proves Proposition 1.3: under slow recurrence (SR), CE, CE2, and TCE are equivalent and invariant under topological conjugacy. It does so by (i) CE+SR ⇒ CE2 via the Graczyk–Smirnov telescope and SR to control the initial loss term (Lemma 2.4), (ii) CE2+SR ⇒ CE using the reversed telescope, Koebe, and that CE2 implies ExpShrink (Lemma 2.6), (iii) ExpShrink+SR ⇒ CE (Lemma 3.3), and (iv) topological invariance via quasiconformal conjugacy once CE is known (Lemma 4.1). These steps appear explicitly in the PDF (Proposition 1.3 and Lemmas 2.4, 2.6, 3.3, 4.1) and match the model’s outline almost point-for-point. Minor discrepancies: the model informally writes the loss exponent as μ(c)−1 whereas the telescope inequality in the paper uses μmax−μ(c), and it attributes a lower bound on shrinking diameters to ExpShrink (which actually gives an upper bound; the needed lower bound is obtained via SR and Koebe). These do not affect the conclusions. Overall, the model and paper use the same chain of known tools (shrinking neighborhoods/telescopes, PRLS03 equivalences, Przytycki’s lemma) to reach the same result. See the paper’s statement and proofs for CE⇒CE2 (Lemma 2.4), CE2⇒CE (Lemma 2.6), ExpShrink⇒CE under SR (Lemma 3.3), and topological invariance (Lemma 4.1) together establishing Proposition 1.3.