Consider the discrete dynamical system Consider the \"tripling map\" on the inte

Question

Consider the discrete dynamical system

Consider the "tripling map" on the interval [0,1]: Draw the graph of T and find the fixed points of T. Explain why T cannot have any attracting periodic points. Show that the periodic points of T are dense in [0,1]. Show that T has sensitive dependence under the initial conditions. More precisely, show that there exists beta > 0 such that for any x0 epsilon [0,1] and any x1 x'o there exists n such that |Tn{x) - Tn(x0) beta. To receive full credit, you have to provide an explicit value for beta and give a careful proof.

Explanation / Answer

i think thismay help :

Exercise T1.12: Consider the 3x mod 1 map of the unit interval [0,1].
Define the distance between a pair of points x, y to be either |x ? y| or
1?|x?y|, whichever is smaller. (We are measuring with the

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