7,259 research outputs found
General concepts of graphs
A little general abstract combinatorial nonsense delivered in this note is a
presentation of some old and basic concepts, central to discrete mathematics,
in terms of new words. The treatment is from a structural and systematic point
of view. This note consists essentially of definitions and summaries.Comment: 20 page
More simple proofs of Sharkovsky's theorem
Recently we have obtained two simple proofs of Sharkovsky's theorem, one with
directed graphs [7] and the other without [8]. In this note, we present yet
more simple proofs of Sharkovsky's theorem.Comment: 5 page
Congruence Identities Arising From Dynamical Systems
By counting the numbers of periodic points of all periods for some interval
maps, we obtain infinitely many new congruence identities in number theory.Comment: 5 page
The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem
Let f(x) be a continuous function from a compact real interval into itself
with a periodic orbit of minimal period m, where m is not an integral power of
2. Then, by Sharkovsky's theorem, for every positive integer n with m \prec n
in the Sharkovsky's ordering defined below, a lower bound on the number of
periodic orbits of f(x) with minimal period n is 1. Could we improve this lower
bound from 1 to some larger number? In this paper, we give a complete answer to
this question.Comment: 11 page
On the Invariance of Li-Yorke Chaos of Interval Maps
In their celebrated "Period three implies chaos" paper, Li and Yorke proved
that if a continuous interval map f has a period 3 point then there is an
uncountable scrambled set S on which f has very complicated dynamics. One
question arises naturally: Can this set S be chosen invariant under f? The
answer is positive for turbulent maps and negative otherwise. In this note, we
shall use symbolic dynamics to achieve our goal. In particular, we obtain that
the tent map T(x) = 1 - |2x-1| on [0, 1] has a dense uncountable invariant
1-scrambled set which consists of transitive points.Comment: 6 page
What make them all so turbulent
We give a unified proof of the existence of turbulence for some classes of
continuous interval maps which include, among other things, maps with periodic
points of odd periods > 1, some maps with dense chain recurrent points and
densely chaotic maps.Comment: 5 pages, 2 figure
On the number of parameters for which the point is a superstable periodic point of
Let be a one-parameter family of real continuous maps
with parameter . For every positive integer , let denote the
number of parameters such that the point is a (superstable)
periodic point of whose least period divides (in particular,
). In this note, we find a recursive way to depict how {\it some}
of these parameters appear in the interval and show that
and this result is generalized
to a class of one-parameter families of continuous real-valued maps that
includes the family .Comment: 7 pages, 1 figur
The lives of period-3 orbits for some quadratic polynomials
In this note, we consider the following two families of quadratic polynomials
and and show that their
respective period-3 orbits live very different lives.Comment: 5 pages, 2 figure
An example of unbounded chaos
Let for all . Then we extend in
the usual way to become a continuous map from the compact topological (but not
metric) space onto itself which also maps the set of irrational
points in onto itself. In this note, we show that (1) on , is topologically mixing, has dense irrational periodic
points, and has topological entropy , where is the
unique positive zero of the polynomial ; (2) has bounded
uncountable {\it invariant} 2-scrambled sets of irrational points in ;
(3) for any countably infinite set of points (rational or irrational) in
, there exists a dense unbounded uncountable {\it invariant}
-scrambled set of irrational transitive points in
such that, for any and any , we have and . This demonstrates the true nature of chaos for .Comment: 10 pages, 2 figure
A Simple Proof of Sharkovsky's Theorem Rerevisited
Based on various strategies and a new general doubling operator, we obtain
several simple proofs of the celebrated Sharkovsky's cycle coexistence theorem.
A simple non-directed graph proof which is especially suitable for a calculus
course right after the introduction of Intermediate Value Theorem is also given
(in section 3).Comment: 28 pages, 5 figures, In this revision, we replace a detailed proof of
(a), (b) and (c) in section 3 and a detailed proof of Sharkovsky's theorem in
section 11. arXiv admin note: substantial text overlap with
arXiv:math/070359
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