Consider the tent map $f : [0, 1] \to [0, 1]$, $$f(x) = 2 x \; \text{if} \; 0 < x < \frac{1}{2}$$ and $$f(x) = 2(1 - x) \; \text{if} \; \frac{1}{2} \leq x \leq 1.$$ What does it mean $\textbf{the limit set of}$ $f$?
Thank you!
Limit set of tent map
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dynamical-systems
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1The expression "limit set" has no meaning in this context. Probably you mean $\omega$-limit set (but not of $f$, instead of a point). – 2017-02-23
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0The context is the dynamic behaviour of iterates of $f$. – 2017-02-23
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1A good reference then (for $\omega$-limit sets): http://web.mat.bham.ac.uk/C.Good/research/pdfs/barwell-etal.pdf. Otherwise you need to be more detailed. – 2017-02-23
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0Since $f$ is surjective its $\omega$-limit set is $[0,1]$. However, this is not much informing about dynamics. The dynamical behavior of this map is illustrated in some detail in the book of R. Devaney: "Introduction to chaotic dynamical systems" if I remember correctly. – 2017-02-23
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0In general, the limit-set of a map is the set of all its limits for different initial values of that map. It looks like the limit-set of this one is for $x=0.5$ i.e. $\{1\}$ – 2017-02-25