TITLE: Alive dynamical system
QUESTION [12 upvotes]: Intuitively, one can say that a dynamical system is alive if one can build a universal Turing machine inside.
So, Conway's Game of Life is alive and shift space should be dead.
I fail to make this definition precise. 
Please let me know if someone made it (or at least was trying to).
(It is  nice if you can make a def. 
BUT I'm mostly interested if you saw such definition in the literature.)
Comments:

There is a relateted question, and it links to this paper by Graça--Campagnolo--Buescu (thanks to Joel).

REPLY [4 votes]: I don't pretend to have an answer to this question, but let me at least try (informally and tentatively) to clarify the distinction. Suppose I have a computational problem and I want to solve it. I also happen to have the Game of Life handy, and plenty of information about it. What I presume the usual statement about the Game of Life encoding a universal Turing machine is that there is some algorithmic way of translating my algorithm (for finding a matching in a bipartite graph, say) into an initial configuration for the Game of Life such that after a certain time I can look at the output and see whether a certain position has a + or a -, and that will agree with what the algorithm gives.
Now suppose that I am presented with the shift space. This time, I run the entire computation, describe this run somehow as a sequence of 0s and 1s, and then apply the shift map repeatedly to the resulting sequence. But what have I gained from applying the shift map? Precisely nothing, because in order to work out the sequence I had to run the entire computation. 
Here perhaps is a genuine difference between the two. Suppose I have a program and I don't know whether it halts, and would like to find out. With the Game of Life, after a finite time I can translate the program into an initial configuration, and there can be some stopping rule, and then I can go off and have lunch and leave the Game of Life chugging away. With the shift space, if it happens that the program doesn't halt, I will never finish the process of creating a sequence on which the shift should act.