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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?

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    It 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

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