Tent Map

Tent shaped graph defined by:

$$f(x)=\{\begin{array}{ll}\mu x& for0\le x\le \frac{1}{2}\\ \mu (1-x)& for\frac{1}{2}\le x\le 1\end{array}$$

In the classic case of $\mu$ parameter of $2$:

Desmos

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Transition Graph

Fully connected transition graph, between $0\to 1$, midline is $\frac{1}{2}$.

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Theorem

The tent map has infinitely many chaotic orbits.

We know $h(x_1)=\ln (2)$. for all $x$ except $\frac{1}{2}$.

So any bounded orbit which avoids $\frac{1}{2}$ and is not Asymptotically Periodic will be chaotic. Derivative of $T^K$ at a period-k point will be $\pm 2^K\to$ all periodic orbits are sources. No sinks. So any AP orbit must EP. e.g.) $x_0=0.{\overline{1011}}_2$ is period-4 orbit.