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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Blokh and M. Lyubich,Attractors and transformations of an interval, Banach Center Publications23 (1986), 427–442.MathSciNetGoogle Scholar
  2. [2]
    A. Blokh and M. Lyubich,On the decomposition of one-dimensional attractors of unimodal maps of the interval, Algebra and Analysis (Leningrad Mathematical Journal)1 (1989), 128–145.MathSciNetGoogle Scholar
  3. [3]
    H. Bruin and J. Hawkins,Exactness and maximal automorphic factors of unimodal maps, Ergodic Theory and Dynamical Systems21 (2001), 1009–1034.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Bruin, S. Luzzatto and S. van Strien,Decay of correlations in one-dimensional dynamics, Annales Scientifiques de l’École Normale Supérieure, to appear.Google Scholar
  5. [5]
    H. Bruin and S. van Strien,Existence of acips for multimodal maps, inGlobal Analysis of Dynamical Systems, Festschrift to Floris Takens for his 60’th birthday, 2001, to appear.Google Scholar
  6. [6]
    L. Carleson and W. Gamelin,Complex Dynamics, Springer, Berlin, 1995.Google Scholar
  7. [7]
    P. Collet and J.-P. Eckmann,Positive Lyapunov exponents and absolute continuity for maps of the interval, Ergodic Theory and Dynamical Systems3 (1983), 13–46.MATHMathSciNetGoogle Scholar
  8. [8]
    J. Graczyk and S. Smirnov,Collet, Eckmann & Hölder, Inventiones Mathematicae133 (1998), 69–96.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. Graczyk and S. Smirnov,Non-uniform hyperbolicity in complex dynamics I, II, Preprint (2001).Google Scholar
  10. [10]
    J. Graczyk and S. Smirnov,Non-uniform hyperbolicity in complex dynamics. I Poincaré series and induced hyperbolicity, Manuscript (2000).Google Scholar
  11. [11]
    M. Lyubich,Ergodic theory for smooth one-dimensional dynamical systems, Preprint, Stony Brook11 (1990).Google Scholar
  12. [12]
    R. Mañé,Hyperbolicity, sinks and measure in one dimensional dynamics, Communications in Mathematical Physics100 (1985), 495–524.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R. Mañé,On a theorem of Fatou, Boletim da Sociedade Brasileira de Matemática (N.S.)24 (1993), 1–11.MATHCrossRefGoogle Scholar
  14. [14]
    W. de Melo and S. van Strien,One-dimensional Dynamics, Springer, Berlin, 1993.MATHGoogle Scholar
  15. [15]
    M. Misiurewicz,Absolutely continuous measures for certain maps of an interval, Publications Mathématiques de l’Institut des Hautes Études Scientifiques53 (1981), 17–51.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    T. Nowicki,Symmetric S-unimodal mappings and positive Liapunov exponents, Ergodic Theory and Dynamical Systems5 (1985), 611–616.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    T. Nowicki and D. Sands,Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Inventiones Mathematicae132 (1998), 633–680.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    T. Nowicki and S. van Strien,Invariant measures under a summability condition for unimodal maps, Inventiones Mathematicae105 (1991), 123–136.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Chr. Pommerenke,Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften299, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  20. [20]
    E. Prado,Ergodicity of conformal measures for unimodal polynomials, Conformal Geometry and Dynamics2 (1998), 29–44.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    F. Przytycki,Iteration of holomorphic Collet-Eckmann maps: conformal and invariant measures, Transactions of the American Mathematical Society350 (1998), 717–742.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    F. Przytycki, J. Rivera-Letelier and S. Smirnov,Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Inventiones Mathematicae151 (2003), 29–63.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    F. Przytycki and S. Rohde,Rigidity of holomorphic Collet-Eckmann repellers, Arkiv för Matematik37 (1999), 357–371.MATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    J. Rivera-Letelier,Rational maps with decay of geometry: Rigidity, Thurston’s algorithm and local connectivity, Preprint, Stony Brook9 (2000).Google Scholar
  25. [25]
    S. van Strien,Transitive maps which are not ergodic with respect to Lebesgue measure, Ergodic Theory and Dynamical Systems16 (1996), 833–848.MATHMathSciNetGoogle Scholar
  26. [26]
    D. Sullivan,Conformal dynamical systems, Lecture Notes in Mathematics1007, Springer, Berlin, 1983, pp. 725–752.Google Scholar

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

Personalised recommendations