Beginners level question : symbolic dynamics and notations

Question

Let $f(.)$ be a chaotic 1 D Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control parameter. So, iteratively, we obtain an array of values $x[1],x[2],\ldots, x[N]$.

Using concepts of symbolic dynamics, if $\mathbf{s} = \{s[0],s[1],\ldots,s[n-1]\}$ is the symbolic dynamics obtained from the following rule: if $x[n] >= 0.5$, then $s[n] = 1$, else $s[n]=0$. Knowing $x[n]$ and $s[n-1]$ is sufficient to determine $x[n-1]$ and so I can write $$x[n-1] = f^{-1}(s[n-1],x[n])$$ where $f^{-1}(.)$ is the inverse of $f(.)$ given a symbol $s[n-1]$. There is a one to one mapping between the symbolic sequence and the initial condition from which the symbolic sequence has been generated. This means that $$x[0] = f^{-1}(x[1]) o f^{-1}(x[1]) o \ldots o f^{-1}(x[n])$$ where $o$ is the functional composition.

Again, in Usama and Zakaria's "Chaos-Based Simultaneous Compression and Encryption for Hadoop"] in Section 2.3.1, the authors apply inverse interval mapping using the inverse function to encode a given a symbolic sequence into an initial condition. A symbolic message can be encoded into an initial condition by reverse interval mapping using the inverse function $f^{-1}(.)$. Then, starting from the initial condition, if the map $f(.)$ is iterated and the symbolic dynamics obtained by state space partition using the intervals computed from the encoding stage, we can obtain the same symbolic sequence.

Let $f(.)$ be the Skew Tent Map and its inverse given below

\begin{equation} \begin{aligned} f^{-1}(I) = x[n] = \begin{cases} p \times I_{s[n]}, \text{symbol} s[n] =0 \\ 1-p \times I_{s[n]}, \text{symbol} s[n] =1 \label{InverseSkewTentMap} \end{cases} \end{aligned} \end{equation} where $I_{1}$ implies interval when the symbol at $s[i]$ is = 0 and $I_{1}$ implies the interval when the symbol at $s[i]$ is = $1$.

The Skew Tent map is expressed as \begin{equation} \begin{aligned} f(x) = \begin{cases} x/p, 0\le x <p \\ (1-x)/(1-p), p \le x \le 1 \label{SkewTentMap} \end{cases} \end{aligned} \end{equation}

The Skew Tent map is related to the symbols by the choice of the partition point $p$.

From concepts of symbolic dynamics, [notations and reading material form the book, Hirsh, Smale and Devaney, "differential equations, dynamical systems and an introduction to chaos" chapter 15 download link https://www.math.upatras.gr/~bountis/files/def-eq.pdf

• if the chaotic map maps numbers from real domain onto itself, $f : \mathcal{R} \rightarrow \mathcal{R}$,
• the shift map defined as, $\sigma: \Sigma_2 \rightarrow \Sigma_2$ where $\Sigma_2$ is the alphabet space for two alphabets 0 and 1 and
• the itenarary map, $S: \mathcal{R} \rightarrow \Sigma_2$ having $S^{-1}: \Sigma_2 \rightarrow \mathcal{R}$

Questions :

Problem : Is the map $f^{-1}$ the same as $S^{-1}$ ?

What is the meaning of conjugacy?

Answer 1

This question is better suited for math.stackexchange.

Usually the inverse of a function $f:A \rightarrow B$ is a relation on $B\times A$, and people sometimes massage things so that they can treat this relation as a function, since they want something taking things from $B$ to $A$ and so that they can write things like $f(f^{-1}(b))=b$, even though $f^{-1}$ may not be a functional relation. (For example because there are two different elements in $A$ such that $f$ of either of them is $b$; which one should I call $f^{-1}(b)$? Many times, both of them.)

Your descriptions above show some massage work, as $B$ "looks like" $A$, but in order to go back, you need some symbol information. Thus as a function, your inverse in your first example is really from $A \times S$ to $A$, and is related to but not actually a functional inverse to $f:A\rightarrow A$, because of the different domain needed. Similarly for the second example, where the relational inverse is expanded to a map on (something like) the power set of $A$ (perhaps twisted a bit) to $A$, whereas often such a map is defined from the power set of $B$ to the power set of $A$. So while some notion of inverting map is indeed going on, it is not the notion one finds in basic set theory.

Nevertheless, we can use set theory to solve your problem. $S^{-1}$ is a relation that (as I infer since it is not described in your post) is between a language and reals (so $B$ is a set of symbol strings) and $f^{-1}$ is a relation involving real numbers which are different from the alphabet, so cannot be the same.

The idea for symbolic dynamics came from a paper of Hadamard on curvature. As I understand it, rather than try to trace the numerical behaviour of the map he was studying on his space, he chose to paint parts of the space with different symbols, and then look at the progression of symbols formed by looking at how the dynamical system took a particular point and sent it through iteration to different parts of the space. The hope was that the behaviour could be understood through this symbolic interpretation, and the numerical analysis could be replaced or at least augmented by a symbolic type of analysis. Thus points in the space near each other that were painted by the same symbol might have the same or similar symbolic trajectories, and this similarity (which is related to but not quite conjugate) would inform or simplify the description of the qualitative behaviour of the map.

There should be many introductory level presentations of symbolic dynamics online. I recommend you do a web search for them to get a better understanding of the subject, especially of conjugacy.

Gerhard "Not A Symbolic Dynamics Expert" Paseman, 2017.04.04.