Asynchronous sliding block maps

We define a notion of asynchronous sliding block map that can be realized b y transducers labeled in A* x B*. Mie show that, under some conditions, it is possible to synchronize this transducer by state splitting, in order to get a transducer which defines the same sliding block map and which is labe led in A x B-k, where k is a constant integer. In the case of a transducer with a strongly connected graph, the synchronization process can be conside red as an implementation of an algorithm of Frougny and Sakarovitch for syn chronization of rational relations of bounded delay. The algorithm can be a pplied in the case where the transducer has a constant integer transmission rate on cycles and has a strongly connected graph. It keeps the locality o f the input automaton of the transducer. We show that the size of the slidi ng window of the synchronous local map grows linearly during the process, h ut that the size of the transducer is intrinsically exponential. In the cas e of non strongly connected graphs, the algorithm of Frougny and Sakarovitc h does not keep the locality of the input automaton of the transducer. We g ive another algorithm to solve this case without losing the good dynamic pr operties that guaranty the state splitting process. AMS Subject Classification. 37B10. 68Q45.