We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

We consider special flows over the rotation on the circle by an irrational α under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that α has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy the weak Ratner property. Moreover, we provide...

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1,2,\cdots ,\infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the $C^{\infty }$ closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...