Consider an infinite binary string $\sigma$ and define its entropy
$$H_1 = -(p_0 \log_2 p_0 + p_1 \log_2 p_1)$$
with $p_i = \lim_{N\rightarrow \infty} N(i)/N$, $N(i)$ the number of $i$'s among the first $N$ digits.
Now consider the very same string as composed of words $w$ of length $n$ and define the corresponding entropies
$$H_n = -\frac{1}{n}\sum_{w \in \lbrace 0,1\rbrace^n} p_w \log_2 p_w $$
A natural candidate for the "true" entropy of $\sigma$ might be
$$H = \lim_{n\rightarrow\infty} H_n$$
Note, that for a truly random string $H_n = H = 1$. For a higly regular string like $0101010\dots$ we have $H_1 = 1, H_n = H = 0$ for $n\geq 2$.
Question 1: Is every value $0 \leq H \leq 1$ possible? Are there sequences which don't reach the limit?
Question 2: Which progressions of $H_n$ are conceivable? Does it have to be a monotonic sequence, or can it oscillate?
Question 3: Under which assumptions does this limit exist?
Now consider finite binary strings $\sigma$ of length $N$. Here we have $H_n = 0$ for $n > N/2$ and $H_{N/2}$ can only take the values $0$ or $1$.
Question 4: What is a natural candidate for the "true" entropy of finite binary strings?