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.

[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.

[3]

R. B. Ash, "Real Analysis and Probability,'', Probability and Mathematical Statistics, (1972).

[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.

[5]

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

[6]

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

[7]

A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps,, Comm. Math. Phys., 199 (1998), 397. 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, (1997).

[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.

[10]

J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, in, 69 (2001), 749.

[11]

B. Cadre and P. Jacob, On pairwise sensitivity,, J. Math. Anal. Appl., 309 (2005), 375. doi: 10.1016/j.jmaa.2005.01.061.

[12]

B. D. Craven, "Lebesgue Measure & Integral,'', Pitman, (1982).

[13]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', The Benjamin/Cummings Publishing Co., (1986).

[14]

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

[15]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067. 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. 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, 1486 (1991), 227.

[18]

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

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. 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, 54 (1995).

[21]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergodic Theory Dynam. Systems, 10 (1990), 717. 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.

[23]

F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergodic Theory Dynam. Systems, 1 (1981), 77. doi: 10.1017/S0143385700001176.

[24]

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

[25]

M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems,, Stony Brook preprint, (1991).

[26]

R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics,, Comm. Math. Phys., 100 (1985), 495.

[27]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,'', Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8 (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 (1993).

[29]

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

[30]

W. Parry, "Entropy and Generators in Ergodic Theory,'', W. A. Benjamin, (1969).

[31]

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

[32]

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

[33]

S. Ruette, Chaos for continuous interval maps. A survey of relationship between the various sorts of chaos,, preprint, (2003).

[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.

[35]

P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982).

[36]

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

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.

[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.

[3]

R. B. Ash, "Real Analysis and Probability,'', Probability and Mathematical Statistics, (1972).

[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.

[5]

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

[6]

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

[7]

A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps,, Comm. Math. Phys., 199 (1998), 397. 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, (1997).

[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.

[10]

J. Buzzi, Thermodynamical formalism for piecewise invertible maps: Absolutely continuous invariant measures as equilibrium states,, in, 69 (2001), 749.

[11]

B. Cadre and P. Jacob, On pairwise sensitivity,, J. Math. Anal. Appl., 309 (2005), 375. doi: 10.1016/j.jmaa.2005.01.061.

[12]

B. D. Craven, "Lebesgue Measure & Integral,'', Pitman, (1982).

[13]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,'', The Benjamin/Cummings Publishing Co., (1986).

[14]

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

[15]

E. Glasner and B. Weiss, Sensitive dependence on initial conditions,, Nonlinearity, 6 (1993), 1067. 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. 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, 1486 (1991), 227.

[18]

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

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. 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, 54 (1995).

[21]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps,, Ergodic Theory Dynam. Systems, 10 (1990), 717. 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.

[23]

F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval,, Ergodic Theory Dynam. Systems, 1 (1981), 77. doi: 10.1017/S0143385700001176.

[24]

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

[25]

M. Lyubich, Ergodic theory for smooth one-dimensional dynamical systems,, Stony Brook preprint, (1991).

[26]

R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics,, Comm. Math. Phys., 100 (1985), 495.

[27]

R. Mañé, "Ergodic Theory and Differentiable Dynamics,'', Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8 (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 (1993).

[29]

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

[30]

W. Parry, "Entropy and Generators in Ergodic Theory,'', W. A. Benjamin, (1969).

[31]

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

[32]

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

[33]

S. Ruette, Chaos for continuous interval maps. A survey of relationship between the various sorts of chaos,, preprint, (2003).

[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.

[35]

P. Walters, "An Introduction to Ergodic Theory,'', Graduate Texts in Mathematics, 79 (1982).

[36]

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

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