Graphes dans les surfaces et ergodicit\'e topologique

The simplest way to make a dynamical system out of a finite connected graph $G$ is to give it a polarization, that is to say a cyclic ordering of the edges incident to a vertex, for each vertex. The phase space $\mathcal{P}(G)$ then consists of all pairs $(v,e)$ where $v$ is a vertex and $e$ is an edge incident to $v$. Such an initial condition gives a position and a momentum. The data $(v,e)$ is of course equivalent to an edge endowed with an orientation $e_{\mathcal O}$. With the polarization, each initial data leads to a leftward walk defined by turning left at each vertex, or making a rebound if there is no other edge. A leftward walk is called complete if it goes through all edges of $G$, not necessarily in both directions. As usual, we define the valence of a vertex as the number of edges incident to it, and we define the valence of a graph as the average of the valences of its vertices. In this article, we prove that if a graph which is embedded in a closed oriented surface of genus $g$ admits a complete leftward walk, then its valence is at most $1 + \sqrt{6g+1}$. We prove furthermore that this result is sharp for infinitely many genera $g$, and that it is asymptotically optimal as $g \to + \infty$. This leads to obstructions for the embeddability of graphs on a surface in a way which admits a complete leftward walk. Since checking that a polarized graph admits a complete leftward walk or not is done in time $4N$, where $N$ is the cardinality of the edges, this obstruction is particularly efficient in terms of computability. This problem has its origins in interesting consequences for what we will call here the topological ergodicity of conservative systems, especially Hamiltonian systems $H$ in two dimensions where the existence of a complete leftward walk corresponds to a topologically ergodic orbit of the system, i.e. an orbit of $H$ visiting all the topology of the surface.