# 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 - A, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
##### References:
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Misiurewicz, Wild attractors of polymodal negative Schwarzian maps,, Comm. Math. Phys., 199 (1998), 397.  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, (1997).   Google Scholar [9] H. Bruin, G. Keller and M. St. Pierre, Adding machines and wild attractors,, Ergodic Theory Dynam. Systems, 17 (1997), 1267.  doi: 10.1017/S0143385797086392.  Google Scholar [10] J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, in, 69 (2001), 749.   Google Scholar [11] B. Cadre and P. Jacob, On pairwise sensitivity,, J. Math. Anal. Appl., 309 (2005), 375.  doi: 10.1016/j.jmaa.2005.01.061.  Google Scholar [12] B. D. Craven, "Lebesgue Measure & Integral,'', Pitman, (1982).   Google Scholar [13] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', The Benjamin/Cummings Publishing Co., (1986).   Google Scholar [14] E. I. Dinaburg, A correlation between topological entropy and metric entropy,, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19.   Google Scholar [15] E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067.  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.  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, 1486 (1991), 227.   Google Scholar [18] S. D. Johnson, Singular measures without restrictive intervals,, Comm. Math. Phys., 110 (1987), 185.  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.  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, 54 (1995).   Google Scholar [21] G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergodic Theory Dynam. Systems, 10 (1990), 717.  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.   Google Scholar [23] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergodic Theory Dynam. Systems, 1 (1981), 77.  doi: 10.1017/S0143385700001176.  Google Scholar [24] E. N. Lorenz, The predictability of hydrodynamic flow,, Trans. New York Acad. Sci., 25 (1963), 409.   Google Scholar [25] M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems,, Stony Brook preprint, (1991).   Google Scholar [26] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics,, Comm. Math. Phys., 100 (1985), 495.   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 (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 (1993).   Google Scholar [29] M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 167.   Google Scholar [30] W. Parry, "Entropy and Generators in Ergodic Theory,'', W. A. Benjamin, (1969).   Google Scholar [31] V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499.   Google Scholar [32] D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Soc. Brasil. Mat., 9 (1978), 83.   Google Scholar [33] S. Ruette, Chaos for continuous interval maps. A survey of relationship between the various sorts of chaos,, preprint, (2003).   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.  doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar [35] P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982).   Google Scholar [36] H. Whitney, On totally differentiable and smooth functions,, Pacific J. Math., 1 (1951), 143.   Google Scholar

show all references

##### References:
 [1] C. Abraham, G. Biau and B. Cadre, Chaotic properties of mappings on a probability space,, J. Math. Anal. Appl., 266 (2002), 420.  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.  doi: 10.1016/j.jmaa.2003.10.029.  Google Scholar [3] R. B. Ash, "Real Analysis and Probability,'', Probability and Mathematical Statistics, (1972).   Google Scholar [4] Y. Baba, I. Kubo and Y. Takahashi, Li-Yorke's scrambled sets have measure $0$,, Nonlinear Anal., 26 (1996), 1611.  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.  doi: 10.2307/27641863.  Google Scholar [6] A. M. Blokh, Sensitive mappings of an interval,, Uspekhi Mat. Nauk, 37 (1982), 189.  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.  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, (1997).   Google Scholar [9] H. Bruin, G. Keller and M. St. Pierre, Adding machines and wild attractors,, Ergodic Theory Dynam. Systems, 17 (1997), 1267.  doi: 10.1017/S0143385797086392.  Google Scholar [10] J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, in, 69 (2001), 749.   Google Scholar [11] B. Cadre and P. Jacob, On pairwise sensitivity,, J. Math. Anal. Appl., 309 (2005), 375.  doi: 10.1016/j.jmaa.2005.01.061.  Google Scholar [12] B. D. Craven, "Lebesgue Measure & Integral,'', Pitman, (1982).   Google Scholar [13] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', The Benjamin/Cummings Publishing Co., (1986).   Google Scholar [14] E. I. Dinaburg, A correlation between topological entropy and metric entropy,, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19.   Google Scholar [15] E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067.  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.  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, 1486 (1991), 227.   Google Scholar [18] S. D. Johnson, Singular measures without restrictive intervals,, Comm. Math. Phys., 110 (1987), 185.  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.  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, 54 (1995).   Google Scholar [21] G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergodic Theory Dynam. Systems, 10 (1990), 717.  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.   Google Scholar [23] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergodic Theory Dynam. Systems, 1 (1981), 77.  doi: 10.1017/S0143385700001176.  Google Scholar [24] E. N. Lorenz, The predictability of hydrodynamic flow,, Trans. New York Acad. Sci., 25 (1963), 409.   Google Scholar [25] M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems,, Stony Brook preprint, (1991).   Google Scholar [26] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics,, Comm. Math. Phys., 100 (1985), 495.   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 (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 (1993).   Google Scholar [29] M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 167.   Google Scholar [30] W. Parry, "Entropy and Generators in Ergodic Theory,'', W. A. Benjamin, (1969).   Google Scholar [31] V. A. Rohlin, Exact endomorphisms of a Lebesgue space,, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499.   Google Scholar [32] D. Ruelle, An inequality for the entropy of differentiable maps,, Bol. Soc. Brasil. Mat., 9 (1978), 83.   Google Scholar [33] S. Ruette, Chaos for continuous interval maps. A survey of relationship between the various sorts of chaos,, preprint, (2003).   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.  doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar [35] P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982).   Google Scholar [36] H. Whitney, On totally differentiable and smooth functions,, Pacific J. Math., 1 (1951), 143.   Google Scholar
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