# 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:
 [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

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
 [1] Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 [2] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [3] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [4] Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223 [5] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [6] Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009 [7] Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215 [8] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [9] Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008 [10] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [11] Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140 [12] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [13] Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451 [14] Boris Kalinin, Anatole Katok, Federico Rodriguez Hertz. Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data". Journal of Modern Dynamics, 2010, 4 (1) : 207-209. doi: 10.3934/jmd.2010.4.207 [15] Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192 [16] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [17] Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367 [18] Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469 [19] Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1 [20] Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

2018 Impact Factor: 1.143