On cusps and flat tops  [ Singularités et points critiques plats ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 571-605.

La théorie de Pesin est développée pour une classe d’applications de l’intervalle, lisses par morceaux. On n’exclut ni des singularités de la dérivée, ni que les points critiques soit plats. On prend comme hypothèse que la dérivée satisfasse à une condition liée à celle de la régularité Hölder.

Nos résultats s’appliquent à des transformations de l’intervalle de classe C 1+ϵ . Comme conséquence, on démontre l’absence de mesure de probabilité invariante et absolument continue par rapport à la mesure de Lebesgue, lorsque les points critiques sont trop plats. Cela étend un résultat de Benedicks et Misiurewicz.

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.

DOI : https://doi.org/10.5802/aif.2858
Classification : 37E05,  37D25
Mots clés : exposant de Lyapunov, théorie de Pesin, mesures invariantes et absolument continues, dynamique sur l’intervalle, points critiques plats.
@article{AIF_2014__64_2_571_0,
     author = {Dobbs, Neil},
     title = {On cusps and flat tops},
     journal = {Annales de l'Institut Fourier},
     pages = {571--605},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {2},
     year = {2014},
     doi = {10.5802/aif.2858},
     mrnumber = {3330915},
     zbl = {06387285},
     language = {en},
     url = {archive.numdam.org/item/AIF_2014__64_2_571_0/}
}
Dobbs, Neil. On cusps and flat tops. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 571-605. doi : 10.5802/aif.2858. http://archive.numdam.org/item/AIF_2014__64_2_571_0/

[1] Araújo, Vítor; Luzzatto, Stefano; Viana, Marcelo Invariant measures for interval maps with critical points and singularities, Adv. Math., Volume 221 (2009) no. 5, pp. 1428-1444 | Article | MR 2522425 | Zbl 1184.37032

[2] Aspenberg, Magnus Rational Misiurewicz maps are rare, Comm. Math. Phys., Volume 291 (2009) no. 3, pp. 645-658 | Article | MR 2534788 | Zbl 1185.37103

[3] Benedicks, Michael; Misiurewicz, Michał Absolutely continuous invariant measures for maps with flat tops, Inst. Hautes Études Sci. Publ. Math. (1989) no. 69, pp. 203-213 | Article | EuDML 104051 | Numdam | MR 1019965 | Zbl 0703.58030

[4] Blokh, A. M.; Lyubich, M. Yu. Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4), Volume 24 (1991) no. 5, pp. 545-573 | EuDML 82305 | Numdam | MR 1132757 | Zbl 0790.58024

[5] Bruin, H. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., Volume 168 (1995) no. 3, pp. 571-580 | Article | MR 1328254 | Zbl 0827.58015

[6] Bruin, H.; Rivera-Letelier, J.; Shen, W.; van Strien, S. Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., Volume 172 (2008) no. 3, pp. 509-533 | Article | MR 2393079 | Zbl 1138.37019

[7] Bruin, Henk; Shen, Weixiao; van Strien, Sebastian Invariant measures exist without a growth condition, Comm. Math. Phys., Volume 241 (2003) no. 2-3, pp. 287-306 | MR 2013801 | Zbl 1098.37034

[8] Bruin, Henk; Todd, Mike Equilibrium states for interval maps: the potential -tlog|Df|, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 4, pp. 559-600 | EuDML 272190 | Numdam | MR 2568876 | Zbl 1192.37051

[9] Díaz-Ordaz, K.; Holland, M. P.; Luzzatto, S. Statistical properties of one-dimensional maps with critical points and singularities, Stoch. Dyn., Volume 6 (2006) no. 4, pp. 423-458 | Article | MR 2285510 | Zbl 1130.37362

[10] Dobbs, Neil Critical points, cusps and induced expansion in dimension one (2006) (Ph. D. Thesis)

[11] Dobbs, Neil Visible measures of maximal entropy in dimension one, Bull. Lond. Math. Soc., Volume 39 (2007) no. 3, pp. 366-376 | Article | MR 2331563 | Zbl 1132.37017

[12] Dobbs, Neil Measures with positive Lyapunov exponent and conformal measures in rational dynamics, Trans. Amer. Math. Soc., Volume 364 (2012) no. 6, pp. 2803-2824 | Article | MR 2888229 | Zbl 1267.37042

[13] Dobbs, Neil; Skorulski, Bartłomiej Non-existence of absolutely continuous invariant probabilities for exponential maps, Fund. Math., Volume 198 (2008) no. 3, pp. 283-287 | Article | MR 2391016 | Zbl 1167.37024

[14] Graczyk, Jacek; Sands, Duncan; Świątek, Grzegorz Metric attractors for smooth unimodal maps, Ann. of Math. (2), Volume 159 (2004) no. 2, pp. 725-740 | Article | MR 2081438 | Zbl 1055.37041

[15] Hofbauer, Franz On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., Volume 34 (1979) no. 3, p. 213-237 (1980) | Article | MR 570882 | Zbl 0422.28015

[16] Hofbauer, Franz On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II, Israel J. Math., Volume 38 (1981) no. 1-2, pp. 107-115 | Article | MR 599481 | Zbl 0456.28006

[17] Hofbauer, Franz; Raith, Peter The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull., Volume 35 (1992) no. 1, pp. 84-98 | Article | MR 1157469 | Zbl 0701.28005

[18] Hofbauer, Franz; Raith, Peter The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval, Canad. Math. Bull., Volume 35 (1992) no. 1, pp. 84-98 | Article | MR 1157469 | Zbl 0701.28005

[19] Keller, Gerhard Lifting measures to Markov extensions, Monatsh. Math., Volume 108 (1989) no. 2-3, pp. 183-200 | Article | EuDML 178444 | MR 1026617 | Zbl 0712.28008

[20] Keller, Gerhard Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, Volume 10 (1990) no. 4, pp. 717-744 | Article | MR 1091423 | Zbl 0715.58020

[21] Ledrappier, François Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynamical Systems, Volume 1 (1981) no. 1, pp. 77-93 | Article | MR 627788 | Zbl 0487.28015

[22] Ledrappier, François Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., Volume 299 (1984) no. 1, pp. 37-40 | MR 756305 | Zbl 0567.58016

[23] Luzzatto, Stefano; Tucker, Warwick Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Hautes Études Sci. Publ. Math. (1999) no. 89, p. 179-226 (2000) | Article | EuDML 104158 | Numdam | MR 1793416 | Zbl 0978.37029

[24] Martens, Marco Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems, Volume 14 (1994) no. 2, pp. 331-349 | Article | MR 1279474 | Zbl 0809.58026

[25] de Melo, Welington; van Strien, Sebastian One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Volume 25, Springer-Verlag, Berlin, 1993, pp. xiv+605 | MR 1239171 | Zbl 0791.58003

[26] Newhouse, Sheldon E. Entropy and volume, Ergodic Theory Dynam. Systems, Volume 8 * (1988) no. Charles Conley Memorial Issue, pp. 283-299 | Article | MR 967642 | Zbl 0638.58016

[27] Parry, William Topics in ergodic theory, Cambridge Tracts in Mathematics, Volume 75, Cambridge University Press, Cambridge, 1981, pp. x+110 | MR 614142 | Zbl 0449.28016

[28] Rohlin, V. A. Exact endomorphisms of a Lebesgue space, Amer. Math. Soc. Transl. (2), Volume 39 (1964), pp. 1-36 | MR 228654 | Zbl 0154.15703

[29] Ruelle, David An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., Volume 9 (1978) no. 1, pp. 83-87 | Article | MR 516310 | Zbl 0432.58013

[30] Rychlik, Marek Bounded variation and invariant measures, Studia Math., Volume 76 (1983) no. 1, pp. 69-80 | EuDML 218512 | MR 728198 | Zbl 0575.28011

[31] Sands, Duncan Misiurewicz maps are rare, Comm. Math. Phys., Volume 197 (1998) no. 1, pp. 109-129 | Article | MR 1646471 | Zbl 0921.58015

[32] Stefano, Luzzatto; Marcelo, Viana Positive Lyapunov exponents for Lorenz-like families with criticalities, Astérisque (2000) no. 261, pp. xiii, 201-237 | MR 1755442 | Zbl 0944.37025

[33] Thunberg, Hans Positive exponent in families with flat critical point, Ergodic Theory Dynam. Systems, Volume 19 (1999) no. 3, pp. 767-807 | Article | MR 1695920 | Zbl 0966.37011

[34] Zweimüller, Roland S-unimodal Misiurewicz maps with flat critical points, Fund. Math., Volume 181 (2004) no. 1, pp. 1-25 | Article | MR 2071693 | Zbl 1065.28009