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 languages (so $A$ and $B$ are both sets 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 it 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.

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.

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 languages (so $A$ and $B$ are both sets 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 it 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.

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 languages (so $A$ and $B$ are both sets 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 it 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.