Sensitive Dependence for Baker's Function

[McCoy13]

Homework Statement

$$B_{\mu}(x) = 2\mu x : 0 \leq x \leq ^{1}/_{2}$$
$$B_{\mu}(x) = \mu (2x-1) : ^{1}/_{2} < x \leq 1$$

Determine the values of $\mu$ for which $B_{\mu}$ has sensitive dependence on initial conditions.

The text defines sensitive dependence on initial conditions as a property of a function provided that for any x in the domain, there is an exists an $\epsilon$ such that for any y in the domain arbitrarily close to x, there exists an n such that the nth iterate of y is more than $\epsilon$ away from the nth iterate of x.

Homework Equations

N/A

The Attempt at a Solution

I was able to determine that $\mu>^{1}/_{2}$ on account of the fact that for values less than this, all the iterates will converge to 0, or if $\mu=^{1}/_{2}$ the iterates are all eventually fixed points and so the distance between any two points will stay exactly the same.

I am having trouble coming up with a scheme for analyzing the case of $^{1}/_{2}<\mu <1$. I have tried to think of points that will eventually have iterates greater than 1/2 and try to separate two points, one below 1/4mu and one above 1/2, but I can't think of how to show (or if it's even true) that such a separation occurs for two points arbitrarily close. I've also tried to think of points that display some kind of eventual predictable behavior, so that if we start with an arbitrary x, we can pick a y arbitrarily close that will eventually do something predictable, and then we can compare how far an eventually periodic y would be from some arbitrary x. I so far have not been able to come up with such a scheme, as I am fairly certain that all periodic points of this function will be repelling, and that the function has period-n points for all n (though I haven't checked this). It is obvious that this implies that the function would display chaotic behavior, but I am having a difficult time thinking of how to show this from the definition given as part of the problem statement.

 

[McCoy13]

I think I figured it out. The phrasing of the definition of sensitive dependence is important because ε may not depend on δ, but n may depend on δ. Therefore it's just a matter of finding a formula for n for a given ε and δ. That will be simple to do because the function simply doubles the distance between the two points with each iteration.