Exercises for Chaos Under Control

Chapter 4: Some Dust in the Tent

40.[A] For s > 2 we have seen the interval [0,1] contains many points escaping to -infinity. Show the points not escaping to -infinity form a Cantor set, and find the similarity dimension of this Cantor set. Note: the dimension will be expressed in terms of the slope s. Answer

41.[A] Let s = 4 in the tent map. To what value does x = 1/4 map? To what value does x = 1/16 map? To what value does x = 1/64 map? Now, suppose x0 = 1/4n. What is x1? What is x2? What are xn-1, xn, and xn+1? Answer

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