dynamical systems and (perhaps) algebraic geometry

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2 total messages Started by Pascal Koiran Tue, 08 Sep 1992 11:07

#58

Author: Pascal Koiran
Date: Tue, 08 Sep 1992 11:07

18 lines
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Hello,

Can someone provide a solution to the following problem, or give references
to related work in the litterature ?

Let f:[0,1]^n-->[0,1]^n (n>1) be a function of the form
f(x)=(P_1(x),P_2(x),...,P_n(x)) where the P_i's are polynomials.
Let's consider the iterates of f. Is the following situation possible ?

for all k, there exists an infinite number of k-cycles (by k-cycle, I mean
a cycle of length exactly k).

Generalize to rational functions instead of polynomials, and to analytic 
functions.

Thanks,

Pascal.

Re: dynamical systems and (perhaps) algebraic geometry

#60

Author: Arthur Rubin
Date: Tue, 08 Sep 1992 19:22

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In <Bu9Ax6.H59@ens-lyon.fr> koiran@ens.ens-lyon.fr (Pascal Koiran) writes:

>Hello,

>Can someone provide a solution to the following problem, or give references
>to related work in the litterature ?

>Let f:[0,1]^n-->[0,1]^n (n>1) be a function of the form
>f(x)=(P_1(x),P_2(x),...,P_n(x)) where the P_i's are polynomials.
>Let's consider the iterates of f. Is the following situation possible ?

>for all k, there exists an infinite number of k-cycles (by k-cycle, I mean
>a cycle of length exactly k).

P_1(x) = g(x_1) = 4 x_1 (1-x_1); P_i(x) = x_i (2<=i<=n) should work, as:

g has k-cycles for all k (well-known result; e.g. (1-cos(2 pi/(2^k-1)))/2
is in a k-cycle of g).

If a_1,a_2,...,a_k is a k-cycle of g, then (a_1,x_2,...,x_n),
(a_2,x_2,...,x_n), ..., (a_k,x_2,...,x_n) is a k-cycle of f.



--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
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