Non-uniformly expanding dynamics in maps with singularities and criticalities

Stephano Luzzatto; Warwick Tucker

Publications Mathématiques de l'IHÉS (1999)

  • Volume: 89, page 179-226
  • ISSN: 0073-8301

How to cite

top

Luzzatto, Stephano, and Tucker, Warwick. "Non-uniformly expanding dynamics in maps with singularities and criticalities." Publications Mathématiques de l'IHÉS 89 (1999): 179-226. <http://eudml.org/doc/104158>.

@article{Luzzatto1999,
author = {Luzzatto, Stephano, Tucker, Warwick},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Lyapunov exponent; Lorenz system; return map; bounded recurrence; binding period; interval maps},
language = {eng},
pages = {179-226},
publisher = {Institut des Hautes Études Scientifiques},
title = {Non-uniformly expanding dynamics in maps with singularities and criticalities},
url = {http://eudml.org/doc/104158},
volume = {89},
year = {1999},
}

TY - JOUR
AU - Luzzatto, Stephano
AU - Tucker, Warwick
TI - Non-uniformly expanding dynamics in maps with singularities and criticalities
JO - Publications Mathématiques de l'IHÉS
PY - 1999
PB - Institut des Hautes Études Scientifiques
VL - 89
SP - 179
EP - 226
LA - eng
KW - Lyapunov exponent; Lorenz system; return map; bounded recurrence; binding period; interval maps
UR - http://eudml.org/doc/104158
ER -

References

top
  1. [ABS77] V. S. AFRAIMOVICH, V. V. BYKOV and L. P. SHIL'NIKOV, On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR 234 (1977), 336-339. Zbl0451.76052
  2. [BC85] M. BENEDICKS and L. CARLESON, On iterations of 1 - ax2 on (- 1, 1), Ann. of Math. 122 (1985), 1-25. Zbl0597.58016MR87c:58058
  3. [BC91] M. BENEDICKS and L. CARLESON, The dynamics of the Hénon map, Ann. of Math. 133 (1991), 73-169. Zbl0724.58042MR92d:58116
  4. [Cos98] M. J. COSTA, Global strange attractors after collision of horseshoes with periodic sinks, PhD thesis, IMPA (1998). 
  5. [dMvS93] W. de MELO and S. VAN STRIEN, One-dimensional dynamics, Springer Verlag, Berlin, 1993. Zbl0791.58003MR95a:58035
  6. [GW79] J. GUCKENHEIMER and R. F. WILLIAMS, Structural stability of Lorenz attractors, Publ. Math. IHES 50 (1979), 59-72. Zbl0436.58018MR82b:58055a
  7. [Hén76] M. HÉNON, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69-77. Zbl0576.58018MR54 #10917
  8. [HL] M. HOLLAND and S. LUZZATTO, Hyperbolicity and statistical properties of two-dimensional maps with criticalities and singularities, work in progress. 
  9. [HP76] M. HÉNON and Y. POMEAU, Two strange attractors with a simple structure, Lect. Notes in Math. 565 (1976), 29-68. Zbl0379.76049MR56 #6742
  10. [Jak81] M. JAKOBSON, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981), 39-88. Zbl0497.58017MR83j:58070
  11. [Lor63] E. N. LORENZ, Deterministic nonperiodic flow, J. Atmosph. Sci. 20 (1963), 130-141. 
  12. [Luz00] S. LUZZATTO, Bounded recurrence of critical points and Jakobson's theorem, London Mathematical Society Lecture Notes 274 (2000). Zbl1062.37027MR2001i:37056
  13. [LV] S. LUZZATTO and M. VIANA, Lorenz-like attractors without continuous invariant foliations, in preparation. 
  14. [LV00] S. LUZZATTO and M. VIANA, Positive Lyapunov exponents for Lorenz-like maps with criticalities, Asterisque, 261 (2000), 201-237. Zbl0944.37025MR2001f:37036
  15. [MV93] L. MORA and M. VIANA, Abundance of strange attractors, Acta Math. 171 (1993), 1-71. Zbl0815.58016MR94k:58089
  16. [PRV] M. J. PACIFICO, A. ROVELLA, and M. VIANA, Persistence of global spiraling attractors, in preparation. 
  17. [PRV98] M. J. PACIFICO, A. ROVELLA, and M. VIANA, Infinite-modal maps with global chaotic behaviour, Ann. of Math. 148 (1998), 1-44. Zbl0916.58029MR99k:58060
  18. [Ree86] M. REES, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup., 4e Série 19 (1986), 383-407. Zbl0611.58038MR88g:58100
  19. [Rov93] A. ROVELLA, The dynamics of perturbations of the contracting Lorenz attractor, Bull. Braz. Math. Soc. 24 (1993), 233-259. Zbl0797.58051MR95a:58097
  20. [Ryc88] M. RYCHLIK, Another proof of Jakobson's theorem and related results, Erg. Th. &amp; Dyn. Syst. 8 (1988), 93-109. Zbl0671.58019MR90b:58149
  21. [Spa82] C. SPARROW, The Lorenz equations : bifurcations, chaos and strange attractors, volume 41 of Applied Mathematical Sciences, Springer Verlag, Berlin, 1982. Zbl0504.58001MR84b:58072
  22. [Thu98] H. THUNBERG, Positive Lyapunov exponents for maps with flat critical points, Erg. Th. &amp; Dyn. Syst. 19 (1998), 767-807. Zbl0966.37011MR2000m:37051
  23. [Tsu93a] M. TSUJII, Positive Lyapunov exponents in families of one-dimensional dynamical systems, Inventiones Mathematicae 111 (1993), 113-137. Zbl0787.58029MR93j:58081
  24. [Tsu93b] M. TSUJII, A proof of Benedicks-Carleson-Jakobson Theorem, Tokyo J. Math. 16 (1993), 295-310. Zbl0801.58027MR94j:58104
  25. [Tuc99] W. TUCKER, The Lorenz attractor exists, C. R. Acad. Sci. Paris t. 328, Série I (1999), 1197-1202. Zbl0935.34050MR2001b:37051
  26. [Yoc] J.-C. YOCCOZ, Weakly Hyperbolic Dynamics, Birkhauser, in preparation. 
  27. [You98] L.-S. YOUNG, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. 147 (1998) 585-650. Zbl0945.37009MR99h:58140

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.