mathematical physics - Can binary sequences generated from ergodic maps be chaotic?

Briefly, the way symbols are generated is:

Consider a one-dimensional chaotic map $T: [0,1]→[0,1]$ and a time series $\{x_n\}_{n=1}^N$ generated with this map. Define a threshold $A$ and a thresholding function $π$:

$$\pi(x) := \begin{cases} 1 &\text{if} \quad x>A \\ 0 & \text{else}\end{cases}$$

Consider a symbolic sequence $\{s_n\}_{n=1}^N$ obtained by applying $π$ to the elements of $\{x_n\}_{n=1}^N$, i.e., $s_i := π(x_i)$

For example, let $A =0.5$ be the threshold and the time series $x = [0.1,\, 0.56,\, 0.6,\, ...]$. Then

$\pi(x_1) = 0$;

$\pi(x_1) = 1$;

$\pi(x_2) = 1$.

Thus, the symbolic representation is $s = [0,1,1]$

I am facing technical difficulties in following the paper. My question is: Such binary sequences are just i.i.d. random variables of some distribution. How can we say that these sequences $s$ are chaotic?

Answer

When such 0/1 sequences are generated, they are just i.i.d random variables of some distribution.

No, they aren’t.

As for identically distributed, consider any sequence when changing $A$. The higher $A$, the higher the probability that the symbol is $1$.

As for independently, consider the sequence generated by a tent map and then transform all values via

$$f(x) := \left(x-\tfrac{1}{5}\right) \bmod 1.$$

The corresponding sequence could as well have been generated by a somewhat shifted tent map, which looks like this:

If you select $A=\tfrac{7}{10}$ (see plot), it should become obvious that the probability that a $1$ is followed by another $1$ is higher than the probability that that a $0$ is followed by a $1$, so the symbol sequence has some memory.

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How can we say that these sequences S are chaotic?

It all boils down to your definition of chaoticity at the end, but consider the following: Let $T$ be the classical tent map (forget the above shift) and let $A:=\tfrac{1}{2}$. Now, if we represent numbers in binary, the effect of the map on a number can be understood as removing its first digit after the decimal point, e.g.,

$$T(0.10100101011101) = 0.0100101011101,$$ $$T(0.0100101011101) = 0.100101011101,$$ and so on. Due to our threshold selection, we also have that $s_i$ is the digit removed from $x_i$ by $T$. Thus your sequence $s$ is the sequence of digits of $x_0$.

Thus, the more precisely, we know $x_0$, the longer can we accurately predict the sequence $s$, but any finite precision will make it impossible for us to predict $s$ forever. This is very reminiscent of the Butterfly Effect.