Stronger Forms of Sensitivity for Measure-Preserving Maps and Semiflows on Probability Spaces

Abstract

This paper is concerned with some stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces. A new form of sensitivity is introduced, called ergodic sensitivity. It is shown that, on a metric probability space with a fully supported measure, if a measure-preserving map is weak mixing, then it is ergodically sensitive and multisensitive; and if it is strong mixing, then it is cofinitely sensitive, where it is not required that the map is continuous and the space is compact. Similar results for measure-preserving semiflows are obtained, where it is required in a result about ergodic sensitivity that the space is compact in some sense and the semiflow is continuous. In addition, relationships between some sensitive properties of a map and its iterations are discussed, including syndetic sensitivity, cofinite sensitivity, ergodic sensitivity as well as usual sensitivity, $n$-sensitivity, and multisensitivity. Moreover, it is shown that multisensitivity, cofinite sensitivity, and ergodic sensitivity can be lifted up by a semiopen factor map.