From the book: M.Amigó, Permutation Complexity in Dynamical Systems, Springer Verlag, 2010 and Based on my readings, a concept of symbolic dynamics exists www2.acqs.org/mathstat/personal_pages/williams/wilshort.pdf) which is represented by the dyadic shift map.
It is the study of the coarse-grain partition/ finite discrete representation of an orbit. I am facing problem in understanding part of this concept which says that the Dyadic map represents the dynamics of the symbolic dynamics.
Question1: In essence, does this mean that any sequence of 0/1 can be generated or modeled by it? How can Dyadic map be used to generate binary random variables when input is also binary?
Question2: Can any chaotic map be used to model/represent binary random variables?
For example: Let a chaotic map $f(x) = 4x(1-x)$ and the starting seed value be x[0] = 0.1. Then, the orbit of x[0] = 0.1, 0.36, 0.921 6, 0.289 01, 0.821 94, 0.585 42, ...
which translates into a random sequence = 0, 0, 1, 0, 1, 1, ... if a threshold of 0.5 is used to binarize the real values of the orbit.
Then there is the dyadic map: $f(x) = 2*x \mod(1)$. When $x$ is a binary bit, then will the output f(x) be binary as well if the dyadic map is the shift map on the space of 0/1 sequences?
How to prove that any chaotic map can be used to represent or model 0/1?
1) You mix the notions strongly. What you mean by random is completely unclear.
You can take every binary sequence of 0,1 and act on it by the dyadic map (or by the corresponding shift, to be more precise). If you take a random real number of the interval [0,1], it (its binary representation) gives you a random binary sequence -- you don't have to use dynamics for that.
What you may ask in this context is whether every sequence of outcomes of the Bernoulli experiment (tossing of an independent random fair coin) corresponds to some of the trajectories of the deterministic dynamical system (shift on the space of binary sequences), and vice versa. Then the answer is "yes" (see the third sentence of my answer).
2) No in any sense. In some cases like $x\to 3 x (\mod 1)$ you work with triadic (ternary) numbers. But there are many examples (and the logistic maps is one of them) where you either can not code the dynamics by a finite alphabet at all, or this coding has bad properties (the conjugacy may not be even Holder continuous. Or may be not continuous at all).