The escape trichotomy for singularly perturbed rational maps

On the dynamics of the McMullen family 𝑅(𝑧)=𝑧^{𝑚}+𝜆/𝑧^{ℓ}

“…Remark. The different cases (n ≤ 1), (n = 2), and (n ≥ 3) constitute the escape trichotomy (see [10]). Note, however, that escape time in [10] is measured for the critical values rather than the critical points.…”

“…The different cases (n ≤ 1), (n = 2), and (n ≥ 3) constitute the escape trichotomy (see [10]). Note, however, that escape time in [10] is measured for the critical values rather than the critical points. A Sierpiński curve is a compact, connected, locally connected, and nowhere dense subset of the plane such that the complementary components (Jordan domains) do not touch, i.e., any two distinct complementary components have disjoint closures.…”

“…It was, and is, the purpose of this paper to give an adequate, though by no means complete answer to this question. This project, however, has to bear in mind that a lot of detailed information is available in a series of recent papers and preprintswhich I realised only in the winter of 2005 ( [2], [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [26]), due to R. Devaney and collaborators, so that the picture to be drawn here has to be complementary to the picture drawn in the above papers; certain overlaps, however, are unavoidable.…”

Proof of a folklore Julia set connectedness theorem and connections with elliptic functions

“…The techniques used in these papers, along with studies done on the maps of the form (1), studied copiously by Devaney and other authors (for example, [3,[6][7][8]) lead to a natural proof of this theorem. Since the Weierstrass elliptic ℘ function has a lot of classical and beautiful identities which aid in the understanding of the dynamics (used in [11] and later papers), this idea leads to many identities for the maps in the family given in (1).…”

Any counterexample to Makienko’s conjecture is an indecomposable continuum

“…In this case, there are uncountably components of the Julia set which do not intersect the boundary of any Fatou component. Later, John Milnor and Tan Lei [ML93] and Devaney, Look, and Uminsky [DLU05] exhibited rational functions with Julia sets homeomorphic to the Sierpinski carpet. It is well-known that the residual Julia set is then a connected dense G δ subset.…”