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Misiurewicz, Michal

Absolutely continuous measures for certain maps of an interval. (English) Zbl 0477.58020

Publ. Math., Inst. Hautes Étud. Sci. 53, 17-51 (1981).
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Cited in 14 Reviews
Cited in 105 Documents

MSC:

37A99 Ergodic theory
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28D20 Entropy and other invariants
54C70 Entropy in general topology

Keywords:

piecewise smooth map of an interval into itself with negative Schwarzian derivative and polynomial behaviour in the neighbourhoods of exceptional points; no periodic attracting points; existence of a probability invariant measure absolutely continuous with respect to the Lebesgue measure; metric entropies; topological entropy
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\textit{M. Misiurewicz}, Publ. Math., Inst. Hautes Étud. Sci. 53, 17--51 (1981; Zbl 0477.58020)
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References:

[1] N. Bourbaki,Fonctions d’une variable réelle (Livre IV), Paris, Hermann, 1958 (chap. 1, § 4, exerc. 1a).
[2] J. Guckenheimer,Sensitive dependence to initial conditions for one dimensional maps, preprint I.H.E.S. (1979). · Zbl 0429.58012
[3] M. Jakobson, Topological and metric properties of one-dimensional endomorphisms,Dokl. Akad. Nauk SSSR,243 (1978), 866–869 (in Russian).
[4] J. Milnor, W. Thurston,On iterated maps of the interval, preprint. · Zbl 0664.58015
[5] M. Misiurewicz, Structure of mappings of an interval with zero entropy,Publ. Math. I.H.E.S.,53 (1981), 000-000.
[6] M. Misiurewicz, W. Szlenk, Entropy of piecewise monotone mappings,Astérisque,50 (1977), 299–310 (full version will appear inStudia Math.,67). · Zbl 0376.54019
[7] W. Parry,Entropy and generators in ergodic theory, New York, Benjamin, 1969. · Zbl 0175.34001
[8] W. Parry, Symbolic dynamics and transformations of the unit interval,Trans. Amer. Math. Soc.,122 (1966), 368–378. · Zbl 0146.18604
[9] R. Shaw,Strange attractors, chaotic behavior and information flow, preprint, Santa Cruz (1978).
[10] D. Singer, Stable orbits and bifurcation of maps of the interval,SIAM J. Appl. Math.,35 (1978), 260–267. · Zbl 0391.58014
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