# American Institute of Mathematical Sciences

February  2012, 32(2): 433-466. doi: 10.3934/dcds.2012.32.433

## On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps

 1 Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain, Spain

Received  September 2010 Revised  July 2011 Published  September 2011

Let $f:I=[0,1]\rightarrow I$ be a Borel measurable map and let $\mu$ be a probability measure on the Borel subsets of $I$. We consider three standard ways to cope with the idea of observable chaos'' for $f$ with respect to the measure $\mu$: $h_\mu(f)>0$ ---when $\mu$ is invariant---, $\mu(L^+(f))>0$ ---when $\mu$ is absolutely continuous with respect to the Lebesgue measure---, and $\mu(S^\mu(f))>0$. Here $h_\mu(f)$, $L^+(f)$ and $S^\mu(f)$ denote, respectively, the metric entropy of $f$, the set of points with positive Lyapunov exponent, and the set of sensitive points to initial conditions with respect to $\mu$.
It is well known that if $h_\mu(f)>0$ or $\mu(L^+(f))>0$, then $\mu(S^\mu(f))>0$, and that (when $\mu$ is invariant and absolutely continuous) $h_\mu(f)>0$ and $\mu(L^+(f))>0$ are equivalent properties. However, the available proofs in the literature require substantially stronger hypotheses than those strictly necessary. In this paper we revisit these notions and show that the above-mentioned results remain true in, essentially, the most general (reasonable) settings. In particular, we improve some previous results from [2], [6], and [23].
Citation: Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
##### References:
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Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems, 1 (1981), 77-93. doi: 10.1017/S0143385700001176.  Google Scholar [24] E. N. Lorenz, The predictability of hydrodynamic flow, Trans. New York Acad. Sci., Ser. 2, 25 (1963), 409-432. Google Scholar [25] M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems, Stony Brook preprint, 1991/11, arXiv:math/9201286. Google Scholar [26] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100 (1985), 495-524, Erratum in Comm. Math. Phys., 112 (1987), 721-724.  Google Scholar [27] R. 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A survey of relationship between the various sorts of chaos, preprint, Université Paris-Sud, 2003. Available from: http://www.math.u-psud.fr/~ruette/publications.html. Google Scholar [34] S. van Strien and E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc., 17 (2004), 749-782. Erratum in J. Amer. Math. Soc., 20 (2007), 267-268. doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar [35] P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [36] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math., 1 (1951), 143-159.  Google Scholar

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##### References:
 [1] C. Abraham, G. Biau and B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431. doi: 10.1006/jmaa.2001.7754.  Google Scholar [2] C. Abraham, G. Biau and B. Cadre, On Lyapunov exponent and sensitivity, J. Math. Anal. Appl., 290 (2004), 395-404. doi: 10.1016/j.jmaa.2003.10.029.  Google Scholar [3] R. B. Ash, "Real Analysis and Probability,'' Probability and Mathematical Statistics, No. 11, Academic Press, New York-London, 1972.  Google Scholar [4] Y. Baba, I. Kubo and Y. Takahashi, Li-Yorke's scrambled sets have measure $0$, Nonlinear Anal., 26 (1996), 1611-1612. doi: 10.1016/0362-546X(95)00044-V.  Google Scholar [5] A. Barrio Blaya and V. Jiménez López, Is trivial dynamics that trivial?, Amer. Math. Monthly, 113 (2006), 109-133. doi: 10.2307/27641863.  Google Scholar [6] A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk, 37 (1982), 189-190. doi: 10.1070/RM1982v037n02ABEH003915.  Google Scholar [7] A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys., 199 (1998), 397-416. doi: 10.1007/s002200050506.  Google Scholar [8] A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,'' Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997.  Google Scholar [9] H. Bruin, G. Keller and M. St. Pierre, Adding machines and wild attractors, Ergodic Theory Dynam. Systems, 17 (1997), 1267-1287. doi: 10.1017/S0143385797086392.  Google Scholar [10] J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states, in "Smooth Ergodic Theory and its Applications" (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001), 749-783.  Google Scholar [11] B. Cadre and P. Jacob, On pairwise sensitivity, J. Math. Anal. Appl., 309 (2005), 375-382. doi: 10.1016/j.jmaa.2005.01.061.  Google Scholar [12] B. D. Craven, "Lebesgue Measure & Integral,'' Pitman, Boston, MA, 1982.  Google Scholar [13] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.  Google Scholar [14] E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.  Google Scholar [15] E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. doi: 10.1088/0951-7715/6/6/014.  Google Scholar [16] J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160. doi: 10.1007/BF01982351.  Google Scholar [17] F. Hofbauer, An inequality for the Ljapunov exponent of an ergodic invariant measure for a piecewise monotonic map of the interval, in "Lyapunov Exponents" (Oberwolfach, 1990), Lecture Notes in Math., 1486, Springer, Berlin, (1991), 227-231.  Google Scholar [18] S. D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 185-190. doi: 10.1007/BF01207362.  Google Scholar [19] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173. doi: 10.1007/BF02684777.  Google Scholar [20] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar [21] G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10 (1990), 717-744. doi: 10.1017/S0143385700005861.  Google Scholar [22] A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces (Russian), Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861-864.  Google Scholar [23] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems, 1 (1981), 77-93. doi: 10.1017/S0143385700001176.  Google Scholar [24] E. N. Lorenz, The predictability of hydrodynamic flow, Trans. New York Acad. Sci., Ser. 2, 25 (1963), 409-432. Google Scholar [25] M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems, Stony Brook preprint, 1991/11, arXiv:math/9201286. Google Scholar [26] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100 (1985), 495-524, Erratum in Comm. Math. Phys., 112 (1987), 721-724.  Google Scholar [27] R. Mañé, "Ergodic Theory and Differentiable Dynamics,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.  Google Scholar [28] W. de Melo and S. van Strien, "One-Dimensional Dynamics,'' Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25, Springer-Verlag, Berlin, 1993.  Google Scholar [29] M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 167-169.  Google Scholar [30] W. Parry, "Entropy and Generators in Ergodic Theory,'' W. A. Benjamin, Inc., New York-Amsterdam, 1969.  Google Scholar [31] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.  Google Scholar [32] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.  Google Scholar [33] S. Ruette, Chaos for continuous interval maps. A survey of relationship between the various sorts of chaos, preprint, Université Paris-Sud, 2003. Available from: http://www.math.u-psud.fr/~ruette/publications.html. Google Scholar [34] S. van Strien and E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc., 17 (2004), 749-782. Erratum in J. Amer. Math. Soc., 20 (2007), 267-268. doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar [35] P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [36] H. Whitney, On totally differentiable and smooth functions, Pacific J. Math., 1 (1951), 143-159.  Google Scholar
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