Book Chapter10.1007/BFB0082847
On iterated maps of the interval
John Milnor,William P. Thurston +1 more
- 01 Jan 1988
- pp 465-563
907
TL;DR: In this paper, an effective calculus for describing the qualitative behavior of the successive iterates of a piecewise monotone mapping is presented, where each iteration has a local minimum or maximum.
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Abstract: Introduction. Mappings from an interval to itself provide the simplest possible examples of smooth dynamical systems. Such mappings have been widely studied in recent years since they occur in quite varied applications, and since their surprisingly complex behavior may provide a useful introduction to the study of higher dimensional situations. The present paper sets up an effective calculus for describing the qualitative behavior of the successive iterates of a piecewise monotone mapping. Let I be a closed interval of real numbers. By definition, a continuous mapping f from I to itself is p!ecewise monotone if I can be subdivided into finitely many subintervals I I, .... I Z on which f is alternately strictly increasing or strictly decreasing. Each such maximal interval on which f is monotone is called a lap of f, and £ = Z(f) points c I ..... cz_ I at which f called the turninq points of f. is the lap number. The separating has a local minimum or maximum are
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References
•Journal Article
Periodic Points of Maps of the Disk and Interval
R. Bowen,J.M. Franks +1 more
TL;DR: In this article, it was shown that the Kupka-smale embedding of Theorem 2 is obtained as a limit of such maps, and that this result cannot be strengthened along the lines of the Theorem 1.
93
Applications conservant une mesure absolument continue par rapport à $dx$ sur $[0,1]$
TL;DR: In this article, sufficient conditions are given such that a differentiable, non invertible, mapg:[0, 1]↦[0, 2] leaves invariant a measure absolutely continuous with respect to the Lebesgue measure.
On Smooth Mappings of the Circle Into Itself
TL;DR: In this article, the authors constructed the set, open and everywhere dense in, of -stable mappings, which is totally disconnected and is topologically conjugate to the topological Markov chain with a finite number of states.
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Periodic Points of Continuous Functions
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The qualitative analysis of a difference equation of population growth.
Steve Smale,R. F. Williams +1 more
TL;DR: Complete qualitative information is obtained for the parameter value b=3.83 for the difference equation fb(x)=b x(1−x) as well as other parameters of the same type.
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