Help in understanding a conjugacy problem

Question

I am studying the book

Applied Symbolic Dynamics and Chaos
 By Bai-lin Hao, Wei-Mou Zheng

The basic premise of the concept of Symbolic Dynamics is :

"Symbolic dynamics may be defined as a “coarse-grained” description of the evolution of a dynamical system. The idea is to partition the state space and to associate a symbol to each partition. Then, a trajectory of the dynamical system can be analyzed as a symbolic sequence. In the case of the Bernoulli shift map, the state space is represented by the invariant interval I=[0,1].

In the case of the Bernoulli shift map (with N=2), a Markov partition may be selected by simply splitting the interval I=[0, 1] with respect to the critical point c=0.5 and, correspondingly, we define the two subintervals $I_0=[0, 0.5)$ and $I_1=[0.5, 1)$. In order to obtain a symbolic description of the dynamics of the chaotic map under consideration, we associate the binary symbol “0” to the subinterval $I_0$ and the symbol “1” to the subinterval $I_1$. Then, the evolution of the state of the Bernoulli map can be described in terms of a symbolic sequence $S={010010 . . . }$."

A symbolic orbit is obtained by writing down the sequence of symbols corresponding to the successive partition elements visited by the point in its orbit. One can learn much about the dynamics of the system by studying its symbolic orbits.

Let the numerical orbit be generated by a Tent Map. According to Topological Conjugacy, Bernoulli Shift Map is a conjugate to Logistic and the Tent Map. Again, in the link Pseudo chaotic communication method using symbolic dynamics under Eq(3) it is mentioned that the symbolic dynamics is described by the Bernoulli map.

In Chapter 2 of the book, there is a mathematical Equation (2.16) and onwards where the inverse of a function/map is used. The purpose is to show that there is a correspondence between the numerical orbit and the symbolized orbit. I am facing difficulty in understanding this statement and the connection with topological congugacy between Tent and Bernoulli Map.

My Questions are :

1. Does the proof in the book essentially mean that the symbolic representation = the numerical orbit ? i.e., numerical orbit of Tent Map = symbolic orbit derived from Bernoulli Map?

2. Can somebody please explain in intuitive non-mathematical way what the Proof in Chapter 2, which is how there is a connection between the numeric and the symbolic orbit is and by symbolic orbit which map are we referring to?

Answer 1

The answer to the first question is No, the numerical and symbolic sequences are different. Several different numerical sequences may share the same symbolic dynamics, but different symbolic sequences must obviously correspond to different numerical orbits.

The numerical orbit $x_{n+1} = f(x_n)$ of the Tent map or other unimodal maps is a real valued sequence, whereas its associated symbolic sequence in this case only takes the values Left or Right (L, R). Values of $x_n$ to the left of the peak of $f(x)$ are labeled L and values to the right are labeled R (or any other pair of symbols).

Now it's easy to establish the connection from $x_n$ to its symbolic sequence (just follow the labeling scheme). The opposite direction is less obvious: can we find a numerical sequence from a symbolic sequence? I'm not going to try to answer this, I'll just give a little hint.

If you try to apply the map $f(x)$ backwards, you'd get a sequence $$ x_0 = f^{-n}(x_n)$$ though this doesn't make sense since each point $x_n$ has two pre-images such that $f(x_{n-1}) = x_n$. But if you label the two branches of the inverse function by $f_L$ and $f_R$, you can express the sequence as a function composition, e.g. $$ x_0 = f^{-1}_L\circ f^{-1}_R \circ f^{-1}_L (x_3).$$ Now you can read out the symbolic sequence from this function composition.