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


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.


Periodic Orbit Real Case Critical Time Periodic Point Periodic Attractor 
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© 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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