The large deviations theorem and sensitivity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f is a topologically strongly ergodic map, then f is sensitively dependent on initial conditions. Moreover, we investigate the relationships between the large deviations theorem and sensitivity, and show that if f satisfies the large deviations theorem, then f is sensitively dependent on initial conditions if and only if f is neither minimal nor equicontinuous.