Statistical properties of unimodal maps

Artur Avila; Carlos Gustavo Moreira

Publications Mathématiques de l'IHÉS (2005)

  • Volume: 101, page 1-67
  • ISSN: 0073-8301

Abstract

top
We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.

How to cite

top

Avila, Artur, and Moreira, Carlos Gustavo. "Statistical properties of unimodal maps." Publications Mathématiques de l'IHÉS 101 (2005): 1-67. <http://eudml.org/doc/104209>.

@article{Avila2005,
abstract = {We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.},
author = {Avila, Artur, Moreira, Carlos Gustavo},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {typical analytic unimodal maps; chaotic attractors; Lyapunov exponents; periodic orbits; critical orbit; physical measures},
language = {eng},
pages = {1-67},
publisher = {Springer},
title = {Statistical properties of unimodal maps},
url = {http://eudml.org/doc/104209},
volume = {101},
year = {2005},
}

TY - JOUR
AU - Avila, Artur
AU - Moreira, Carlos Gustavo
TI - Statistical properties of unimodal maps
JO - Publications Mathématiques de l'IHÉS
PY - 2005
PB - Springer
VL - 101
SP - 1
EP - 67
AB - We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.
LA - eng
KW - typical analytic unimodal maps; chaotic attractors; Lyapunov exponents; periodic orbits; critical orbit; physical measures
UR - http://eudml.org/doc/104209
ER -

References

top
  1. 1. V. Arnold, Dynamical systems, in Development of mathematics 1950–2000, pp. 33–61, Birkhäuser, Basel 2000. Zbl0963.37003MR1796837
  2. 2. A. Avila, M. Lyubich, W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps. Invent. Math., 154 (2003), 451–550. Zbl1050.37018MR2018784
  3. 3. A. Avila, C. G. Moreira, Statistical properties of unimodal maps: the quadratic family. Ann. Math., 161 (2005), 827–877. Zbl1078.37029MR2153401
  4. 4. A. Avila, C. G. Moreira, Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative. Geometric methods in dynamics. I. Astérisque, 286 (2003), 81–118. Zbl1046.37021MR2052298
  5. 5. A. Avila, C. G. Moreira, Phase-Parameter relation and sharp statistical properties for general families of unimodal maps, preprint (http://www.arXiv.org), to appear in Contemp. Math., volume on “Geometry and Dynamics”, ed. by E. Ghys, J. Eells, M. Lyubich, J. Palis, J. Seade. Zbl1145.37022
  6. 6. M. Benedicks, L. Carleson, On iterations of 1-ax 2 on (-1,1). Ann. Math., 122 (1985), 1–25. Zbl0597.58016MR799250
  7. 7. A. M. Blokh, M. Yu. Lyubich, Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Éc. Norm. Supér., IV. Sér., 24 (1991), 545–573. Zbl0790.58024MR1132757
  8. 8. M. Jacobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys., 81 (1981), 39–88. Zbl0497.58017MR630331
  9. 9. G. Keller, T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps. Commun. Math. Phys., 149 (1992), 31–69. Zbl0763.58024MR1182410
  10. 10. O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition. Ann. Math., 152 (2000), 743–762. Zbl0988.37044MR1815700
  11. 11. O. S. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the Ck topology. Ann. Math., 157 (2003), 1–43. Zbl1215.37022MR1954263
  12. 12. A. N. Livsic, The homology of dynamical systems. Usp. Mat. Nauk, 27 (1972), no. 3(165), 203–204. MR394768
  13. 13. M. Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. Math., 140 (1994), 347–404. Note on the geometry of generalized parabolic towers. Manuscript (2000) (http://www.arXiv.org). Zbl0821.58014MR1298717
  14. 14. M. Lyubich, Dynamics of quadratic polynomials, I–II. Acta Math., 178 (1997), 185–297. Zbl0908.58053MR1459261
  15. 15. M. Lyubich, Dynamics of quadratic polynomials, III. Parapuzzle and SBR measure. Astérisque, 261 (2000), 173–200. Zbl1044.37038MR1755441
  16. 16. M. Lyubich, Feigenbaum-Coullet-Tresser Universality and Milnor’s Hairiness Conjecture. Ann. Math., 149 (1999), 319–420. Zbl0945.37012
  17. 17. M. Lyubich, Almost every real quadratic map is either regular or stochastic. Ann. Math., 156 (2002), 1–78. Zbl1160.37356MR1935840
  18. 18. R. Mañé, Hyperbolicity, sinks and measures for one-dimensional dynamics. Commun. Math. Phys., 100 (1985), 495–524. Zbl0583.58016MR806250
  19. 19. M. Martens, W. de Melo, The multipliers of periodic points in one-dimensional dynamics, Nonlinearity, 12 (1999), 217–227. Zbl0989.37032MR1677736
  20. 20. W. de Melo, S. van Strien, One-dimensional dynamics. Springer 1993. Zbl0791.58003MR1239171
  21. 21. J. Milnor, Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intell., 19 (1997), 30–32. Zbl0883.28004
  22. 22. J. Milnor, W. Thurston, On iterated maps of the interval, Dynamical Systems, Proc. U. Md., 1986–87, ed. by J. Alexander. Lect. Notes Math., 1342 (1988), 465–563. Zbl0664.58015MR970571
  23. 23. T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math., 132 (1998), 633–680. Zbl0908.58016MR1625708
  24. 24. D. Ruelle, A. Wilkinson. Absolutely singular dynamical foliations. Commun. Math. Phys., 219 (2001), 481–487. Zbl1031.37029MR1838747
  25. 25. M. Shub, D. Sullivan, Expanding endomorphisms of the circle revisited. Ergodic Theory Dyn. Syst., 5 (1985), 285–289. Zbl0583.58022MR796755
  26. 26. M. Shub, A. Wilkinson, Pathological foliations and removable zero exponents. Invent. Math., 139 (2000), 495–508. Zbl0976.37013MR1738057
  27. 27. M. Tsujii, Positive Lyapunov exponents in families of one dimensional dynamical systems. Invent. Math., 111 (1993), 113–137. Zbl0787.58029MR1193600

NotesEmbed ?

top

You must be logged in to post comments.