A hierarchical dynamical map is proposed as the basic framework for sensory cortical mapping. To show how the hierarchical dynamical map works in cognitive processes, we applied it to a typical cognitive task known as priming, in which cognitive performance is facilitated as a consequence of prior experience. Prior to the priming task, the network memorizes a sensory scene containing multiple objects presented simultaneously using a hierarchical dynamical map. Each object is composed of different sensory features. The hierarchical dynamical map presented here is formed by random itinerancy among limit-cycle attractors into which these objects are encoded. Each limit-cycle attractor contains multiple point attractors into which elemental features belonging to the same object are encoded. When a feature stimulus is presented as a priming cue, the network state is changed from the itinerant state to a limit-cycle attractor relevant to the priming cue. After a short priming period, the network state reverts to the itinerant state. Under application of the test cue, consisting of some feature belonging to the object relevant to the priming cue and fragments of features belonging to others, the network state is changed to a limit-cycle attractor and finally to a point attractor relevant to the target feature. This process is considered as the identification of the target. The model consistently reproduces various observed results for priming processes such as the difference in identification time between cross-modality and within-modality priming tasks, the effect of interval between priming cue and test cue on identification time, the effect of priming duration on the time, and the effect of repetition of the same priming task on neural activity.