Several sufficient conditions for a map and a semi-flow to be ergodically sensitive

ABSTRACT

In this paper, we are concerned solely with the ergodic sensitivity for maps (resp. semi-flows), where these maps and semi-flows may not be continuous (in topology). A few new sufficient conditions under which a map (resp. a semi-flow) on a metric space is ergodically sensitive are presented, where such maps and semi-flows may not be continuous (in topology). In particular, we prove that the topologically strong ergodicity of a measure-preserving map (resp. a measure-preserving semi-flow) on a metric probability space with a fully supported measure implies its ergodical sensitivity.

KEYWORDS:

• Sensitivity
• topologically weak mixing
• topologically strong ergodicity
• ergodic sensitivity
• measure-preserving semi-flow

MR(2000) Subject Classification:

• 37A05
• 37A25
• 54H20
• 28D05

Acknowledgment

The author is very grateful to the referees for their careful reading, comments, and suggestions, which help us improve this paper.

This research was supported by the Project of Enhancing School With Innovation of Guangdong Ocean University (Grant NO. GDOU2016050207) and the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City (Grant 2010C3112005).

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Project of Enhancing School With Innovation of Guangdong Ocean University [grant number GDOU2016050207]; the Key Scientific and Technological Research Project of Science and Technology Department of Zhanjiang City [grant number 2010C3112005].