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On cusps and flat tops. (Singularités et points critiques plats.) (English. French summary) Zbl 1361.37032
Summary: 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 M. Benedicks and M. Misiurewicz [Publ. Math., Inst. Hautes Étud. Sci. 69, 203–213 (1989; Zbl 0703.58030)].

##### MSC:
 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37E05 Dynamical systems involving maps of the interval
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##### References:
 [1] Araújo, Vítor; Luzzatto, Stefano; Viana, Marcelo, Invariant measures for interval maps with critical points and singularities, Adv. Math., 221, 5, 1428-1444, (2009) · Zbl 1184.37032 [2] Aspenberg, Magnus, Rational misiurewicz maps are rare, Comm. Math. Phys., 291, 3, 645-658, (2009) · Zbl 1185.37103 [3] Benedicks, Michael; Misiurewicz, Michał, Absolutely continuous invariant measures for maps with flat tops, Inst. Hautes Études Sci. Publ. Math., 69, 203-213, (1989) · 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), 24, 5, 545-573, (1991) · Zbl 0790.58024 [5] Bruin, H., Induced maps, Markov extensions and invariant measures in one-dimensional dynamics, Comm. Math. Phys., 168, 3, 571-580, (1995) · 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., 172, 3, 509-533, (2008) · Zbl 1138.37019 [7] Bruin, Henk; Shen, Weixiao; van Strien, Sebastian, Invariant measures exist without a growth condition, Comm. Math. Phys., 241, 2-3, 287-306, (2003) · Zbl 1098.37034 [8] Bruin, Henk; Todd, Mike, Equilibrium states for interval maps: the potential $$-t\log \vert Df\vert ,$$ Ann. Sci. Éc. Norm. Supér. (4), 42, 4, 559-600, (2009) · 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., 6, 4, 423-458, (2006) · Zbl 1130.37362 [10] Dobbs, Neil, Critical points, cusps and induced expansion in dimension one, (2006) [11] Dobbs, Neil, Visible measures of maximal entropy in dimension one, Bull. Lond. Math. Soc., 39, 3, 366-376, (2007) · Zbl 1132.37017 [12] Dobbs, Neil, Measures with positive Lyapunov exponent and conformal measures in rational dynamics, Trans. Amer. Math. Soc., 364, 6, 2803-2824, (2012) · Zbl 1267.37042 [13] Dobbs, Neil; Skorulski, Bartłomiej, Non-existence of absolutely continuous invariant probabilities for exponential maps, Fund. Math., 198, 3, 283-287, (2008) · Zbl 1167.37024 [14] Graczyk, Jacek; Sands, Duncan; Świątek, Grzegorz, Metric attractors for smooth unimodal maps, Ann. of Math. (2), 159, 2, 725-740, (2004) · Zbl 1055.37041 [15] Hofbauer, Franz, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., 34, 3, 213-237 (1980), (1979) · Zbl 0422.28015 [16] Hofbauer, Franz, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. II, Israel J. Math., 38, 1-2, 107-115, (1981) · 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., 35, 1, 84-98, (1992) · 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., 35, 1, 84-98, (1992) · Zbl 0701.28005 [19] Keller, Gerhard, Lifting measures to Markov extensions, Monatsh. Math., 108, 2-3, 183-200, (1989) · Zbl 0712.28008 [20] Keller, Gerhard, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10, 4, 717-744, (1990) · Zbl 0715.58020 [21] Ledrappier, François, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynamical Systems, 1, 1, 77-93, (1981) · Zbl 0487.28015 [22] Ledrappier, François, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math., 299, 1, 37-40, (1984) · Zbl 0567.58016 [23] Luzzatto, Stefano; Tucker, Warwick, Non-uniformly expanding dynamics in maps with singularities and criticalities, Inst. Hautes Études Sci. Publ. Math., 89, 179-226 (2000), (1999) · Zbl 0978.37029 [24] Martens, Marco, Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems, 14, 2, 331-349, (1994) · Zbl 0809.58026 [25] de Melo, Welington; van Strien, Sebastian, One-dimensional dynamics, 25, xiv+605 pp., (1993), Springer-Verlag, Berlin · Zbl 0791.58003 [26] Newhouse, Sheldon E., Entropy and volume, Ergodic Theory Dynam. Systems, 8 *, Charles Conley Memorial Issue, 283-299, (1988) · Zbl 0638.58016 [27] Parry, William, Topics in ergodic theory, 75, x+110 pp., (1981), Cambridge University Press, Cambridge · Zbl 0449.28016 [28] Rohlin, V. A., Exact endomorphisms of a Lebesgue space, Amer. Math. Soc. Transl. (2), 39, 1-36, (1964) · Zbl 0154.15703 [29] Ruelle, David, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9, 1, 83-87, (1978) · Zbl 0432.58013 [30] Rychlik, Marek, Bounded variation and invariant measures, Studia Math., 76, 1, 69-80, (1983) · Zbl 0575.28011 [31] Sands, Duncan, Misiurewicz maps are rare, Comm. Math. Phys., 197, 1, 109-129, (1998) · Zbl 0921.58015 [32] Stefano, Luzzatto; Marcelo, Viana, Positive Lyapunov exponents for Lorenz-like families with criticalities, 261, xiii, 201-237, (2000) · Zbl 0944.37025 [33] Thunberg, Hans, Positive exponent in families with flat critical point, Ergodic Theory Dynam. Systems, 19, 3, 767-807, (1999) · Zbl 0966.37011 [34] Zweimüller, Roland $$, S$$-unimodal misiurewicz maps with flat critical points, Fund. Math., 181, 1, 1-25, (2004) · Zbl 1065.28009
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