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Section 2.10 Tent maps and Cantor sets

For \(\lambda \in [0,4]\text{,}\) the logistic map \(g_{\lambda}\) has the nice property that \(g_{\lambda}:[0,1]\to[0,1]\text{.}\) That is, it maps the unit interval into itself. As a result, the dynamics are bounded and we don't really need to worry about escaping orbits. When \(\lambda=2\text{,}\) the logistic map is surjective; it maps \([0,1]\) onto \([0,1]\text{.}\)

In this section, we'll consider what happens when \(\lambda > 4\text{.}\) We'll see there is a bizarre set, called a Cantor set, that forms an invariant set of points that never escapes under iteration of \(g_{\lambda}\text{.}\)

First, we'll study a family of functions called the tent maps that are somewhat simpler. For this family of functions, the prototypical Cantor set, the ternary set, can arise as the invariant set.