Is there an equation that relates minimum time complexity, minimum space complexity, and entropy of the output of a function? It seems to me that there should be a relatively intuitive relationship between the three, something like $E=f(T+S)$ where $E$ is shannon entropy of the output, $T$ is time, $S$ is space, and f is a linear function or something like that. The main argument being that any function can be written as a hash map that takes its input to its output. This hashmap could be compressed based on its entropy.
I have a hunch that the only way to decrease the space an algorithm uses more than that is to increase the time complexity, as if the hashmap is moved onto a time axis instead of a space one. Is this crazy? Has there been any research done on this subject?
Edit for clarification: What I mean by "entropy of the output of a function" is this. Assume a function $g$ takes as input a bit string $x$ and outputs a message $y$. Now $g$ can thought of a source who's $x$th output is $y$. By "entropy of the output" I mean the shannon entropy of $g$ when thought of as a source.
Again, the argument is that $g$ can be implemented using a hash map in exponential space but in $O(1)$ time. However, I have a hunch that it seems reasonable that this space complexity can be reduced based on the entropy of $g$ (i.e. the entropy of the hashmap) and still maintain $O(1)$ time, essentially by compressing the hash map.
Beyond this reduction I also have a hunch that that you can decrease the space further, but in doing so you have to increase time proportionally. The idea being that you can't actually compress the information in the hash map any more than you already have (even when you compress using other measures of entropy like kolmogorov complexity), you can just "move" it along the time axis.