# Expansion of derivatives in one-dimensional dynamics

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## Abstract

We study the expansion of derivatives along orbits of real and complex one-dimensional maps for infinitely many, for uniform constants

*f*, whose Julia set*J*_{f}attracts a finite set*Crit*of non-flat critical points. Assuming that for each*c*ε*Crit*, either |*D f*^{n}(*f*(*c*))|→∞ (if*f*is real) or*b*_{n}·|*Df*^{n}(*f*(*c*))|→∞ for some summable sequence {*b*_{n}} (if*f*is complex; this is equivalent to summability of |*D f*^{n}(*f*(*c*))|^{−1}), we show that for every*x*ε*J*_{f}\U_{ i }*f*^{−i}(*Crit*), there exist*ℓ*(*x*)≤max_{ c }*ℓ*(*c*) and*K′*(*x*)>0$$|Df^n (x)^{l(x)} \ge K^1 (x).\prod\limits_{i = 0}^{s - 1} {(K_i .|} Df^{n_i - n_i + 1} (f(c_i ))|)$$

*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$$D_k (ci) = \left\{ {\begin{array}{*{20}c} {\max (\lambda ,K.|Df^k (f(c_i ))|)} \\ {\max (\lambda ,K.b_k .|Df^k (f(c_i ))|)} \\\end{array}} \right\}\begin{array}{*{20}c} {if f is real,} \\ {if f is complex,} \\\end{array}$$

*K*>0 and λ>1. If all*c*ε*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}=\(\hat C\) or*J*_{f}has zero Lebesgue measure. Also (assuming all critical points have the same order) there exist*k*>0 such that if*n*is the smallest integer such that*x*enters a certain critical neighbourhood, then |*Df*^{n}(*x*)|≥*k*.## Keywords

Periodic Orbit Real Case Critical Time Periodic Point Periodic Attractor
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