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, U d? (x) = 0 if and only if x = 1. By a straightforward calculation, we have 0 < U d? (0) < 1. Now we divide the arguments into two cases

, A d? (1) ? A d? (?) = J d? = ?. If there exists at least one fixed point of U d? in (0, 1), we denote all of them by 0 < x 1 <, particular

, For every d ? 2 and ? ? R, the Julia set J d? is not a Sierpi?sk carpet

, But this contradicts Lemma 5.3. The proofs of Theorems 5.5 and ?? are finished. By computer experiments, it is shown that A d? (1) ? A d? (?) = {z 0 } for ? ? C, Note that if J d? is a Sierpi?ski carpet

, In this section, we divide the parameter space of T d? into the non-escaping locus M d union countably many capture domains. Recall that A d? (1) and A d? (?) are the immediate superattracting basins of 1 and ? respectively

, For each ? ? C * , the following conditions are equivalent: (1) The Julia set J d? of T d? is a quasicircle

?. D?, In particular, ? k ? A d? (1) if and only if ? l ? A d? (1)

, If J d? is a quasicircle, the Fatou set of T d? consists of two simply connected Fatou components A d? (1) and A d? (?) whose common boundary is J d? . Since T d? permutes 1 and ?, by (5.9), it follows that (2) holds and {? 1 , · · · , ? d } lies in a single Fatou component. Applying the Riemann-Hurwitz formula to U d? : A d? (?) ? A d? (?), We first prove

, we have (3) ? (4) ? (5)

=. D?, Note that T d? : A d? (1) ? A d? (?) is d to 1. We claim that ? k ? A d? (1) for every 0 ? k ? d ? 1. In fact, if not, then 1 ? ? has Specifically

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