Abstract: In the 1970s, D. Sullivan discovere a dictionary between the action of Kleinian groups and the iteration of rational maps. In this talk, we first give a brief introduction to Sullivan's dictionary. Then, we focus on the topic of harmonic measures on both sides of Sullivan's dictionary and introduce my work for extending the dictionary. More precisely, we will discuss the construction of a dynamical graph with a class of random walk on it, which produces a generalized harmonic measure for an expanding dynamical system (X,f). We will show the relation between the phase space X and the Martin boundary of the random walk. We will also discuss a formula about the fractal dimension of the harmonic measure and asymptotic quantities.