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Expansion of derivatives in onedimensional dynamics
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UNSPECIFIED (2003) Expansion of derivatives in onedimensional dynamics. ISRAEL JOURNAL OF MATHEMATICS, 137 . pp. 223263.
Research output not available from this repository, contact author.Abstract
We study the expansion of derivatives along orbits of real and complex onedimensional maps f, whose Julia set J(f) attracts a finite set Crit of nonflat 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)\(l(x)) greater than or equal to K'(x)Pi(i=0)(s1)D(nini+1)(c(i+1))
for infinitely many n. Here 0 = n(s) < (. . .) < n(1) < n(0) = n are socalled critical times, c(i) is a point in Crit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows orb(x) for n(i)  n(i+1) iterates, and
Dk(c(i)) = {(max(lambda, K . \Dfk(f(ci))\) 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:  00212172  
Official Date:  2003  
Dates: 


Volume:  137  
Number of Pages:  41  
Page Range:  pp. 223263  
Publication Status:  Published 
Data sourced from Thomson Reuters' Web of Knowledge
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