UNSPECIFIED (2003) Expansion of derivatives in one-dimensional dynamics. ISRAEL JOURNAL OF MATHEMATICS, 137 . pp. 223-263. ISSN 0021-2172

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Abstract

We study the expansion of derivatives along orbits of real and complex one-dimensional maps f, whose Julia set J(f) attracts a finite set Crit of non-flat critical points. Assuming that for each c is an element of Crit, either \Df(n)(f(c))\ --> infinity (if f is real) or b(n) (.) \Df(n)(f(c))\ --> infinity for some summable sequence {b(n)} (if f is complex; this is equivalent to summability of \Df(n)(f(c))\(-1)), we show that for every x is an element of J(f)\boolean OR(i)f(-i)(Crit), there exist l(x) less than or equal to max(c)l(c) and K'(x) > 0 such that \Df(n)(x)\(l(x)) greater than or equal to K'(x)Pi(i=0)(s-1)D(ni-ni+1)(c(i+1)) for infinitely many n. Here 0 = n(s) < (. . .) < n(1) < n(0) = n are so-called critical times, c(i) is a point in Crit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows orb(x) for n(i) - n(i+1) iterates, and D-k(c(i)) = {(max(lambda, K . \Dfk(f(ci))\) if f is real,)(max(lambda, K . bk . \Dfk(f(ci))\) if f is complex,) for uniform constants K > 0 and lambda > 1. If all c is an element of Crit have the same critical order, then K'(x) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, either J(f) = (C) over cap or J(f) has zero Lebesgue measure. Also (assuming all critical points have the same order) there exist kappa > 0 such that if n is the smallest integer such that x enters a certain critical neighbourhood, then \Df(n)(x)\ greater than or equal to kappa.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	ISRAEL JOURNAL OF MATHEMATICS
Publisher:	MAGNES PRESS
ISSN:	0021-2172
Date:	2003
Volume:	137
Number of Pages:	41
Page Range:	pp. 223-263
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/8397

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