On finite limit sets for transformations on the unit interval
TLDR
An infinite sequence of finite or denumerable limit sets is found for a class of many-to-one transformations of the unit interval into itself and the structure and order of occurrence is universal for the class.About:
This article is published in Journal of Combinatorial Theory, Series A.The article was published on 1973-07-01 and is currently open access. It has received 521 citations till now. The article focuses on the topics: Limit superior and limit inferior & Limit (mathematics).read more
Citations
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Symbolic dynamics. II. Bifurcations in billiards and smooth potentials
TL;DR: In this article, the singular bifurcations in a dispersive billiard are discussed in terms of symbolic dynamics and they are compared to an example of a Bifurcation tree in a smooth potential.
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A recursion formula for the number of stable orbits in the cubic map
TL;DR: In this paper, a recursion formula for the number of stable cycles in the cubic map is given by analysing different types of bifurcations and investigating the stability of all periodic orbits.
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Conway Numbers and Iteration Theory
TL;DR: In this article, it was shown that the real numbers can be generated in one-to-one correspondence with certain sequences of positive and negative sequences of integers, and that these sequences can be expressed as fixed points of the trapezodal map.
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Devil's carpet of topological entropy and complexity of global dynamical behavior
TL;DR: In this article, the concept of an equal topological entropy class (ETEC) is established by the dual star products, and all the infinitely many ETEC plateaus and single points are harmonically organized in the kneading parameter plane, they construct a multifractal devil's carpet.
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Supertracks and nth order windows in the chaotic regime
TL;DR: In this article, the main lines of a renormalization theory for supertrack functions are sketched and a scaling relation yielding n th order windows in the chaotic domain for a large class of one-dimensional maps is obtained.
References
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Book
Non-linear transformation studies on electronic computers
P. R. Stein,Stanisław Ulam +1 more
TL;DR: In this paper, the properties of non-linear transformations in Euclidean spaces are examined, and the invariant points, finite sets, and invariant subsets of the transformations and the means for obtaining them constructively are considered.
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Eine Bemerkung über die Gruppe der topologischen Abbildungen der Kreislinie auf sich selbst
Józef Schreier,Stanislaw M. Ulam +1 more
Journal ArticleDOI