General question. Let $\eta_{n}$ be a sequence of invariant measures on $\{0,1,2,...,p-1\}^{\mathbb{N}}$ and $B$ the Bernoulli uniform measure. Knowing that $\eta_{n} \rightarrow B$ in the weak-* topology, this, somehow, could help me to prove that $h(\eta_{n})\rightarrow\log p$?

I know it is easy to produce zero-entropy measures that converge in the weak-* topology to $B$. What I would like to know is: having the weak-* topology more some friendly hypothesis on the measures $\eta_{n}$, could I guarantee growth of entropy?

finitely determinedproperty of a Bernoulli measure $B$ saysif$\eta_n$ converge weakly to $B$and$h(\eta_n)\to h(B)$, then $\bar d(\eta_n,B)\to 0$. This is a key part of Ornstein theory. $\endgroup$ – Anthony Quas Jan 29 '16 at 16:09not truethat weak$^*$-convergence to the uniform Bernoulli measure gives convergence in entropy. For example, here is a construction of a measure: $\eta_n=\frac{1}{np^n}\sum_{w\in\{0,\ldots,p-1\}^n}\sum_{k=0}^{n-1}\delta_{ \sigma ^k(\bar w)}$, where $\bar w$ is the periodic point $(\bar w)_i=w_{i\bmod n}$. Obviously $h(\eta_n)=0$. It's not hard to check that $\eta_n$ converge weakly to $B$. $\endgroup$ – Anthony Quas Jan 29 '16 at 19:36