Jacek Serafin

POSSIBLE ENTROPY FUNCTIONS

We characterize the class of functions which occur as the entropy function defined on the set of invariant measures of a (minimal) topological dynamical system. Namely, these are all non-negative affine functionsh, defined on metrizable Choquet simplices, which are non-decreasing limits of upper semi-continuous functions. Ifh is itself upper semi-continuous then it can be realized as the entropy function in an expansive dynamical system. The constructions are done effectively using minimal almost 1-1 extensions over a rotation of a group ofp-adic integers (in the expansive case, the construction leads to Toeplitz flows).

Code length between Markov processes

LetX1 andX2 be two mixing Markov shifts over finite alphabet. If the entropy ofX1 is strictly larger than the entropy ofX2, then there exists a finitary homomorphism ϕ:X1→X2 such that the code length is anLp random variable for allp<4/3. In particular, the expected length of the code ϕ is finite.

A short proof of the Ornstein theorem

We give a short (and fairly elementary) proof of the ‘residual Sinai theorem’ of which the Ornstein theorem is an immediate consequence. In principle, we follow the lines of [ BKS ] but our proof avoids any substantial quotations, in particular, invoking any characterizations of systems isomorphic to Bernoulli shifts. The core of the proof is five pages long and it relies only on standard facts in ergodic theory.