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, 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 $X$.