Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full entropy point?


By transitive I mean that there is a point with a dense orbit. By minimal I mean that every orbit is dense. By full entropy point I mean the concept defined in [1](https://www.ams.org/journals/tran/2007-359-12/S0002-9947-07-04357-7/S0002-9947-07-04357-7.pdf), that is a point $x$ such that the topological entropy, restricted to any of its closed neighborhoods, coincides with the entropy $h(f)$ of the system.


Finally, for every closed neighbor $K(x)$ of $x$, let $N(\epsilon, n, K)$ be the largest cardinality of an $(n,\epsilon)$-separated subset $A\subset K$ in the metric $d\_n(x,y)=\max\_{i=0,\dots,n}d(f^i(x),f^i(y))$. Then the entropy restricted to $K(x)$, in symbols $h(f,K)$, is defined as the usual limit
$$\lim\_{\epsilon\to 0}\lim\_{n\to\infty} \frac 1n\log{N(\epsilon, n, K)}.$$
Notice that $h(f)=\sup\_K h(f,K)$, where the supremum is taken over all compact subsets of $X$.



Non-minimal system in which every point is a full entropy point
I guess this is not hard to do with a subshift. Take $X$ to be your favorite minimal positive entropy subshift on $\{0,1\}$ (such examples are constructed in Hahn-Katznelson, among other works).


Now, choose any $x \in X$ and define $y$ on $\{0,1,\*\}$ as
$y = .\* \ x\_1 \* x\_2 x\_3 \* x\_4 x\_5 x\_6 \* x\_7 x\_8 x\_9 x\_{10} \* \ldots$


Define $Y$ to be the orbit closure of $y$. Clearly $Y$ is transitive by definition, and is not minimal since it strictly contains $X$. It's not hard to check that $h(Y) = h(X)$; by the ergodic theorem, any ergodic measure on $Y$ gives $\*$ zero measure, and so is supported on $X$. Then $h(Y) = h(X)$ by the variational principle.


Finally, every point of $Y$ should have full entropy, since every point $z \in Y$ contains arbitrarily long words of $X$, and so all words in the language of $X$ by minimality. So if I understand your definition correctly, $z$ should have full entropy.


I don't see how any of this could be carried over to $[0,1]$, but as we discussed in the comments, it may be impossible to construct such an example on the interval.


