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Israel Journal of Mathematics

, Volume 137, Issue 1, pp 223–263 | Cite as

Expansion of derivatives in one-dimensional dynamics

  • Henk Bruin
  • Sebastian van Strien
Article

Abstract

We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ f attracts a finite setCrit of non-flat critical points. Assuming that for eachcεCrit, either |D f n(f(c))|→∞ (iff is real) orb n·|Df n(f(c))|→∞ for some summable sequence {b n} (iff is complex; this is equivalent to summability of |D f n(f(c))|−1), we show that for everyxεJ f\U i f −i(Crit), there exist(x)≤max c (c) andK′(x)>0
$$|Df^n (x)^{l(x)} \ge K^1 (x).\prod\limits_{i = 0}^{s - 1} {(K_i .|} Df^{n_i - n_i + 1} (f(c_i ))|)$$
for infinitely manyn. Here 0=n s<…<n 1<n 0=n are so-called critical times,c i is a point inCrit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows orb(x) forn i−ni +1 iterates, and
$$D_k (ci) = \left\{ {\begin{array}{*{20}c} {\max (\lambda ,K.|Df^k (f(c_i ))|)} \\ {\max (\lambda ,K.b_k .|Df^k (f(c_i ))|)} \\\end{array}} \right\}\begin{array}{*{20}c} {if f is real,} \\ {if f is complex,} \\\end{array}$$
, for uniform constantsK>0 and λ>1. If allcεCrit have the same critical order, thenK′(x) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, eitherJ f=\(\hat C\) orJ f has zero Lebesgue measure. Also (assuming all critical points have the same order) there existk>0 such that ifn is the smallest integer such thatx enters a certain critical neighbourhood, then |Df n(x)|≥k.

Keywords

Periodic Orbit Real Case Critical Time Periodic Point Periodic Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University 2003

Authors and Affiliations

  • Henk Bruin
    • 1
  • Sebastian van Strien
    • 2
  1. 1.Department of MathematicsUniversity of GroningenAV GroningenThe Netherlands
  2. 2.Department of MathematicsUniversity of WarwickCoventryUK

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