Lagmental Vicfred

The Tent Map Is Coded by Binary Itineraries by Vicfred

Last updated: Tue 03 April 2012

tags: mathematics, dynamical systems, chaos, Lyapunov exponents

Recording which side of one half an orbit visits produces symbolic dynamics. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Objects and notation

A discrete dynamical system iterates \(x_{n+1}=F(x_n)\), while a flow solves \(\dot x=V(x)\). Fixed points, periodic orbits, and invariant sets organize long-term behavior.

$$ T(x)=\begin{cases}2x,&0\le x\le\tfrac12,\\2-2x,&\tfrac12\le x\le1,\end{cases} $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \iota(x)=(s_0,s_1,\ldots),\qquad s_n=\mathbf1_{\{T^n(x)\ge1/2\}} $$

Push the symbols

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \iota\circ T=\sigma\circ\iota,\qquad\sigma(s_0,s_1,s_2,\ldots)=(s_1,s_2,\ldots) $$

Structural reading

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad T(x)=\begin{cases}2x,&0\le x\le\tfrac12,\\2-2x,&\tfrac12\le x\le1,\end{cases},\\[5pt] \mathsf{C}\;&:\quad \iota(x)=(s_0,s_1,\ldots),\qquad s_n=\mathbf1_{\{T^n(x)\ge1/2\}}. \end{aligned} $$

A hypothesis worth keeping

Sensitive dependence is not the same as randomness. A deterministic system may be chaotic while remaining exactly specified by its initial condition.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \iota(x)=(s_0,s_1,\ldots),\qquad s_n=\mathbf1_{\{T^n(x)\ge1/2\}} \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Mon 02 January 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.