In this paper we provide a necessary and sufficient condition for the amenability of the automorphism group of Fra\"iss\'e structures and apply it to prove the non-amenability of the automorphism groups of the directed graph $\mathbf{ S}(3)$ and the Boron tree structure $\mathbf{ T}$. Also, we provide a negative answer to the Unique Ergodicity-Generic Point problem of Angel-Kechris-Lyons [AKL]. By considering $\mathrm{ GL}(\mathbf{ V}_\infty)$, where $\mathbf{ V}_\infty$ is the countably infinite dimensional vector space over a finite field $F_q$, we show that the unique invariant measure on the universal minimal flow of $\mathrm{ GL}(\mathbf{ V}_\infty)$ is not supported on the generic orbit.
Categories: math.LO math.CO math.DS math.PR