The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

On cusps and flat tops

Neil Dobbs[1]

  • [1] Department of Mathematics and Statistics PL 68 FIN-00014 University of Helsinki Finland

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 571-605
  • ISSN: 0373-0956

Abstract

top
Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to C 1 + ϵ . The critical points are not required to verify a non-flatness condition, so the results are applicable to C 1 + ϵ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

How to cite

top

Dobbs, Neil. "On cusps and flat tops." Annales de l’institut Fourier 64.2 (2014): 571-605. <http://eudml.org/doc/275668>.

@article{Dobbs2014,
abstract = {Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^\{1+\epsilon \}$. The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^\{1+\epsilon \}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.},
affiliation = {Department of Mathematics and Statistics PL 68 FIN-00014 University of Helsinki Finland},
author = {Dobbs, Neil},
journal = {Annales de l’institut Fourier},
keywords = {Lyapunov exponent; Pesin theory; absolutely continuous invariant measures; interval dynamics; flat critical points},
language = {eng},
number = {2},
pages = {571-605},
publisher = {Association des Annales de l’institut Fourier},
title = {On cusps and flat tops},
url = {http://eudml.org/doc/275668},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Dobbs, Neil
TI - On cusps and flat tops
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 571
EP - 605
AB - Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1+\epsilon }$. The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1+\epsilon }$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.
LA - eng
KW - Lyapunov exponent; Pesin theory; absolutely continuous invariant measures; interval dynamics; flat critical points
UR - http://eudml.org/doc/275668
ER -

References

top
  1. Vítor Araújo, Stefano Luzzatto, Marcelo Viana, Invariant measures for interval maps with critical points and singularities, Adv. Math. 221 (2009), 1428-1444 Zbl1184.37032MR2522425
  2. Magnus Aspenberg, Rational Misiurewicz maps are rare, Comm. Math. Phys. 291 (2009), 645-658 Zbl1185.37103MR2534788
  3. Michael Benedicks, Michał Misiurewicz, Absolutely continuous invariant measures for maps with flat tops, Inst. Hautes Études Sci. Publ. Math. (1989), 203-213 Zbl0703.58030MR1019965
  4. A. M. Blokh, M. Yu. Lyubich, Measurable dynamics of S -unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), 545-573 Zbl0790.58024MR1132757
  5. H. Bruin, Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys. 168 (1995), 571-580 Zbl0827.58015MR1328254
  6. H. Bruin, J. Rivera-Letelier, W. Shen, S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math. 172 (2008), 509-533 Zbl1138.37019MR2393079
  7. Henk Bruin, Weixiao Shen, Sebastian van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys. 241 (2003), 287-306 Zbl1098.37034MR2013801
  8. Henk Bruin, Mike Todd, Equilibrium states for interval maps: the potential - t log | D f | , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 559-600 Zbl1192.37051MR2568876
  9. K. Díaz-Ordaz, M. P. Holland, S. Luzzatto, Statistical properties of one-dimensional maps with critical points and singularities, Stoch. Dyn. 6 (2006), 423-458 Zbl1130.37362MR2285510
  10. Neil Dobbs, Critical points, cusps and induced expansion in dimension one, (2006) 
  11. Neil Dobbs, Visible measures of maximal entropy in dimension one, Bull. Lond. Math. Soc. 39 (2007), 366-376 Zbl1132.37017MR2331563
  12. Neil Dobbs, Measures with positive Lyapunov exponent and conformal measures in rational dynamics, Trans. Amer. Math. Soc. 364 (2012), 2803-2824 Zbl1267.37042MR2888229
  13. Neil Dobbs, Bartłomiej Skorulski, Non-existence of absolutely continuous invariant probabilities for exponential maps, Fund. Math. 198 (2008), 283-287 Zbl1167.37024MR2391016
  14. Jacek Graczyk, Duncan Sands, Grzegorz Świątek, Metric attractors for smooth unimodal maps, Ann. of Math. (2) 159 (2004), 725-740 Zbl1055.37041MR2081438
  15. Franz Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math. 34 (1979), 213-237 (1980) Zbl0422.28015MR570882
  16. Franz Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II, Israel J. Math. 38 (1981), 107-115 Zbl0456.28006MR599481
  17. Franz Hofbauer, Peter Raith, The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull. 35 (1992), 84-98 Zbl0701.28005MR1157469
  18. Franz Hofbauer, Peter Raith, The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull. 35 (1992), 84-98 Zbl0701.28005MR1157469
  19. Gerhard Keller, Lifting measures to Markov extensions, Monatsh. Math. 108 (1989), 183-200 Zbl0712.28008MR1026617
  20. Gerhard Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems 10 (1990), 717-744 Zbl0715.58020MR1091423
  21. François Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynamical Systems 1 (1981), 77-93 Zbl0487.28015MR627788
  22. François Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 37-40 Zbl0567.58016MR756305
  23. Stefano Luzzatto, Warwick Tucker, Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Hautes Études Sci. Publ. Math. (1999), 179-226 (2000) Zbl0978.37029MR1793416
  24. Marco Martens, Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), 331-349 Zbl0809.58026MR1279474
  25. Welington de Melo, Sebastian van Strien, One-dimensional dynamics, 25 (1993), Springer-Verlag, Berlin Zbl0791.58003MR1239171
  26. Sheldon E. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems 8 * (1988), 283-299 Zbl0638.58016MR967642
  27. William Parry, Topics in ergodic theory, 75 (1981), Cambridge University Press, Cambridge Zbl0449.28016MR614142
  28. V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Amer. Math. Soc. Transl. (2) 39 (1964), 1-36 Zbl0154.15703MR228654
  29. David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87 Zbl0432.58013MR516310
  30. Marek Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80 Zbl0575.28011MR728198
  31. Duncan Sands, Misiurewicz maps are rare, Comm. Math. Phys. 197 (1998), 109-129 Zbl0921.58015MR1646471
  32. Luzzatto Stefano, Viana Marcelo, Positive Lyapunov exponents for Lorenz-like families with criticalities, (2000), xiii, 201-237 Zbl0944.37025MR1755442
  33. Hans Thunberg, Positive exponent in families with flat critical point, Ergodic Theory Dynam. Systems 19 (1999), 767-807 Zbl0966.37011MR1695920
  34. Roland Zweimüller, S -unimodal Misiurewicz maps with flat critical points, Fund. Math. 181 (2004), 1-25 Zbl1065.28009MR2071693

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.