Hierarchical Type Classes and Their Entropy Functions

For each j ≥ 1, if T_j is the finite rooted binary tree with 2^j leaves, the hierarchical type class of binary string x of length 2^j is obtained by placing the entries of x as labels on the leaves of T_j and then forming all permutations of x according to the permutations of the leaf labels under all isomorphisms of tree T_j into itself. The set of binary strings of length 2^j is partitioned into hierarchical type classes, and in each such class, all of the strings have the same type (n₀^j,n₁^j), where n₀^j,n₁^j are respectively the numbers of zeroes and ones in the strings. Let p(n₀^j,n₁^j) be the probability vector (n₀^j/2^j, n₁^j/2^j) belonging to the set P₂ of all two-dimensional probability vectors. For each j ≥ 1, and each of the 2^j + 1 possible types (n₀^j,n₁^j), a hierarchical type class S(n₀^j,n₁^j) is specified. Conditions are investigated under which there will exist a function h : P² → [0, ∞) such that for each p ∈ P₂, if {(n₀^j,n₁^j) : j ≥ 1} is any sequence of types for which p(n₀^j,n₁^j) → p, then the sequence {2^-j log₂(card(S(n₀^j,n₁^j))) : j ≥ 1} converges to h(p). Such functions h, called hierarchical entropy functions, play the same role in hierarchical type class coding theory that the Shannon entropy function on P₂ does in traditional type class coding theory, except that there are infinitely many hierarchical entropy functions but only one Shannon entropy function. One of the hierarchical entropy functions h that is studied is a self-affine function for which a closed-form expression is obtained making use of an iterated function system whose attractor is the graph of h.