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 and Continuous Dynamical Systems, 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), 420-431. doi: 10.1006/jmaa.2001.7754.

[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.

[3]

R. B. Ash, "Real Analysis and Probability,'' Probability and Mathematical Statistics, No. 11, Academic Press, New York-London, 1972.

[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.

[5]

A. Barrio Blaya and V. Jiménez López, Is trivial dynamics that trivial?, Amer. Math. Monthly, 113 (2006), 109-133. doi: 10.2307/27641863.

[6]

A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk, 37 (1982), 189-190. doi: 10.1070/RM1982v037n02ABEH003915.

[7]

A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys., 199 (1998), 397-416. doi: 10.1007/s002200050506.

[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.

[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.

[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.

[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.

[12]

B. D. Craven, "Lebesgue Measure & Integral,'' Pitman, Boston, MA, 1982.

[13]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.

[14]

E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.

[15]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. doi: 10.1088/0951-7715/6/6/014.

[16]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160. doi: 10.1007/BF01982351.

[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.

[18]

S. D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 185-190. doi: 10.1007/BF01207362.

[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.

[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.

[21]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10 (1990), 717-744. doi: 10.1017/S0143385700005861.

[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.

[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.

[24]

E. N. Lorenz, The predictability of hydrodynamic flow, Trans. New York Acad. Sci., Ser. 2, 25 (1963), 409-432.

[25]

M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems, Stony Brook preprint, 1991/11, arXiv:math/9201286.

[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.

[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.

[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.

[29]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 167-169.

[30]

W. Parry, "Entropy and Generators in Ergodic Theory,'' W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[31]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.

[32]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.

[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.

[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.

[35]

P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[36]

H. Whitney, On totally differentiable and smooth functions, Pacific J. Math., 1 (1951), 143-159.

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), 420-431. doi: 10.1006/jmaa.2001.7754.

[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.

[3]

R. B. Ash, "Real Analysis and Probability,'' Probability and Mathematical Statistics, No. 11, Academic Press, New York-London, 1972.

[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.

[5]

A. Barrio Blaya and V. Jiménez López, Is trivial dynamics that trivial?, Amer. Math. Monthly, 113 (2006), 109-133. doi: 10.2307/27641863.

[6]

A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk, 37 (1982), 189-190. doi: 10.1070/RM1982v037n02ABEH003915.

[7]

A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys., 199 (1998), 397-416. doi: 10.1007/s002200050506.

[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.

[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.

[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.

[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.

[12]

B. D. Craven, "Lebesgue Measure & Integral,'' Pitman, Boston, MA, 1982.

[13]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'' The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.

[14]

E. I. Dinaburg, A correlation between topological entropy and metric entropy, (Russian) Dokl. Akad. Nauk SSSR, 190 (1970), 19-22.

[15]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions, Nonlinearity, 6 (1993), 1067-1075. doi: 10.1088/0951-7715/6/6/014.

[16]

J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160. doi: 10.1007/BF01982351.

[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.

[18]

S. D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 185-190. doi: 10.1007/BF01207362.

[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.

[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.

[21]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10 (1990), 717-744. doi: 10.1017/S0143385700005861.

[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.

[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.

[24]

E. N. Lorenz, The predictability of hydrodynamic flow, Trans. New York Acad. Sci., Ser. 2, 25 (1963), 409-432.

[25]

M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems, Stony Brook preprint, 1991/11, arXiv:math/9201286.

[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.

[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.

[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.

[29]

M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 167-169.

[30]

W. Parry, "Entropy and Generators in Ergodic Theory,'' W. A. Benjamin, Inc., New York-Amsterdam, 1969.

[31]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.

[32]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.

[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.

[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.

[35]

P. Walters, "An Introduction to Ergodic Theory,'' Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[36]

H. Whitney, On totally differentiable and smooth functions, Pacific J. Math., 1 (1951), 143-159.

[1]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[4]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[5]

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

[6]

Chuchu Chen, Jialin Hong, Yulan Lu. Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022098

[7]

Jonathan C. Mattingly, Etienne Pardoux. Invariant measure selection by noise. An example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4223-4257. doi: 10.3934/dcds.2014.34.4223

[8]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[9]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215

[10]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[11]

Paola Mannucci, Claudio Marchi, Nicoletta Tchou. Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 119-128. doi: 10.3934/dcdss.2019008

[12]

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

[13]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[14]

Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435

[15]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451

[16]

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

[17]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192

[18]

Andriy Stanzhytsky, Oleksandr Misiats, Oleksandr Stanzhytskyi. Invariant measure for neutral stochastic functional differential equations with non-Lipschitz coefficients. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022005

[19]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[20]

Min Qian, Jian-Sheng Xie. Entropy formula for endomorphisms: Relations between entropy, exponents and dimension. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 367-392. doi: 10.3934/dcds.2008.21.367

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]