Gromov’s entropy for sheafs

The entropy  h(S) of a set S is the logarithm of the number of elements in S. Gromov’s remarkable insight (which he elaborated on in a course I attended a decade ago) by taking the logarithm we can ensure additivity: h(S\times T)=h(S)+h(T). Perhaps one should also observe a more obvious property, monotonicity: S\subset T\implies h(S)\leq h(T).

Gromov observed that in place of the logarithm of the cardinality, one can use any other quantity that is additive (and monotonic). Indeed there is another familiar additive (and monotonic) quantity: dimension. By using dimension in this way, Gromov introduced the idea of mean dimension of the space of holomorphic maps, which is analogous to the density of entropy.

Here I will comment on building on Gromov’s idea in a couple of different ways.  The starting point in both cases is that there is an obvious connection to Sheafs.

Sheafs and Subadditivity: Suppose we have a sheaf on a topological space X. This  means that we associate to each open set $U\subset latex X$ an object – here a set S(U), perhaps with the structure of a vector space. If V\subset U, there is a corresponding restriction map from S(U) to S(V). Further, given elements s_U\in S(U) and $s_V\in latex S(V)$ whose restrictions to U\cap V agree, there is a unique element in S(U\cup V) whose restrictions to U and V are s_U and s_V. The obvious examples of sheafs are continuous (smooth, holomorphic) functions on U. Gromov considers sheafs of holomorphic functions with appropirate bounds to ensure finite-dimensionality.

Suppose we are given a sheaf and an appropriate notion of entropy h(S(U) for the sets S(U) with the given structure – for example, logarithm of the cardinality or dimension of vector spaces. We can then associate to open sets U the function H(U)=h(S(U)). Additivity and monotonicity then give:

\min(H(U),H(V)) \leq(H(U\cup V)\leq H(U)+H(V).

Independence and Conditional Entropy: If U\cap V=\phi, then we see that H(U\cup V)=H(U)+H(V), so the sets (or rather the systems with these sets as domains) are independent. We can take this to be  the definition of independence. More generally, we can define the conditional entropy:

H(U\vert V)=H(U\cap V)-H(V)

This satisfies 0\leq H(U\vert V)\leq H(U), with equality in the second inequality characterising independence.

Domains of holomorphy:

We can consider the sheaf of holomorphic functions satisfying some appropriate conditions ensuring finite dimensionality: $latex\frac{\partial f}{\partial z_i}\leq Cf$ seem promising. We can then consider the function H(U)=dim(s(U). We get associated conditional entropies.

If U is contained in the holomorphic convex hull of V, then H(U\vert V)=0. Thus, we get a nice refinement of notions like domains of holomorphy.

Graphs and Colourings:

Given a countable graph, we consider its set of vertices as a discrete set. We can associate to a set U of vertices the set of colourings of the vertices in U using red, blue and green, so that adjacent vertices have different colours. This gives a sheaf, and the entropy may give insights into 3-colourings.


About sgadgil

I am on the faculty of the Department of Mathematics, Indian Institute of Science.
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