Let $f=\lambda x(1-x)$ be a logistic map with $\lambda>4$. How to show that set of all periodic points of $f$ in $\Lambda$ is countable and the set of point in $\Lambda$ with dense orbits is uncountable?
Logistic function
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statistics
dynamical-systems
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0It is a little too long to write the answer, but you can read the book 'Differential Equations, Dynamical Systems, and an Introduction to Chaos', section 15.3-15.6. – 2012-11-07
1 Answers
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A periodic point of $f$ must be among the solutions of $f^{(k)}(x)=x$. [Here the $(k)$ means the $k$th iterate.] Since in your example $f$ is a nondegenerate polynomial of degree 2, all the iterates are polynomials of positive degree at least 2.
So when $x$ is subtracted to obtain the fixed point equation, you stall have degree at least 2, hence only a finite number of periodic points whose period divides k. I think this can convert to a valid argument about the periodic points. For the uncountability of the set with dense orbits, that would follow if one could rule out non-dense nonperiodic case, but that (to me) seems like one needs to know something about how your particular recursion is working.