Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps
Pages: 433 - 466, Issue 2, February 2012

doi:10.3934/dcds.2012.32.433      Abstract        References        Full text (574.0K)           Related Articles

Alejo Barrio Blaya - Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email)
Víctor Jiménez López - Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email)

1 C. Abraham, G. Biau and B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431.       
2 C. Abraham, G. Biau and B. Cadre, On Lyapunov exponent and sensitivity, J. Math. Anal. Appl., 290 (2004), 395-404.       
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.       
5 A. Barrio Blaya and V. Jiménez López, Is trivial dynamics that trivial?, Amer. Math. Monthly, 113 (2006), 109-133.       
6 A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk, 37 (1982), 189-190.       
7 A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys., 199 (1998), 397-416.       
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.       
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.       
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.       
16 J. Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys., 70 (1979), 133-160.       
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.       
19 A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.       
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.       
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.       
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.       
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.       

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