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...y of Julia sets and combinatorial rigidity for unicritical polynomials are proved in [16] and [2] by means of principal nest and modified principal nest. For dynamics of multimodal maps, see [33] and =-=[34]-=-. Let A be an open set and x ∈ A. We denote the connected component of A containing x by Comp x(A). Given an open set X consisting of finitely many puzzle pieces(not necessarily the same depth) such t...

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...composition of f we can associate an extended map F dened on I n = f(x; i) : x 2 I ; 1 i ng (in other words, I n is a disjoint union of n copies of I) by (1) F (x; i) = (f i (x); i + 1 mod n) In [21=-=-=-], we proved that deep renormalizations in innitely renormalizable multimodal maps are multimodal maps of type n. This is the reason to restrict our attention for this kind of map. We say that J is a ...

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...h bounded combinatorial type, satisfies this assumption (Theorem 2.6), and recently it has been proved that this is also the case for other onedimensional maps with finitely many critical points (see =-=[23]-=-). However there is no result of this type for two-dimensional maps. If an (an)n-infinitely renormalizable map f has the property that any two atoms of the same generation are disjoint (we recall that...

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...ch decomposition of f we can associate an extended map F defined on In = {(x, i) : x ∈ I, 1 ≤ i ≤ n} (in other words, In is a disjoint union of n copies of I) by (1) F (x, i) = (fi(x), i+ 1 mod n) In =-=[Sm1]-=-, we proved that deep renormalizations in infinitely renormalizable multimodal maps are multimodal maps of type n. This is the reason to restrict our attention for this kind of map. We say that J is a...

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...x, i), (y, j) ∈ In, we say that (x, i) < (y, j) if i = j and x < y. The intervals of In are the sets J × {i}, for some J ⊂ I and i < n. If ci is the critical point of fi, denote C(F) = {(i, ci)}i. In =-=[Sma]-=-, we proved that deep renormalizations in infinitely renormalizable multimodal maps are multimodal maps of type n. This is the reason to restrict our attention for this kind of map. Definition 1.2. We...

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...orics, were given by Lyubich [15], Lyubich e Yampolsky [16], Graczyk e Swiatek [12], Levin e van Strien [10]. For multimodal analytic maps, infinitely renormalizable with bounded combinatorics Smania =-=[4]-=- proved “complex bounds”. We obtain complex bounds for the first component of the renormalization operator R̃α which is a univalent map. This tool is useful because it allow us to define the complex r...