TY - JOUR

T1 - A note on sensitivity of semigroup actions

AU - Kontorovich, Eduard

AU - Megrelishvili, Michael

PY - 2008/1

Y1 - 2008/1

N2 - It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey (Am. Math. Mon. 99, 332-334, 1992), Silverman (Rocky Mt. J. Math. 22, 353-375, 1992) and Glasner and Weiss (Nonlinearity 6, 1067-1075, 1993). In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner (Ergodic Theory via Joinings, 2003). Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.

AB - It is well known that for a transitive dynamical system (X,f) sensitivity to initial conditions follows from the assumption that the periodic points are dense. This was done by several authors: Banks, Brooks, Cairns, Davis and Stacey (Am. Math. Mon. 99, 332-334, 1992), Silverman (Rocky Mt. J. Math. 22, 353-375, 1992) and Glasner and Weiss (Nonlinearity 6, 1067-1075, 1993). In the latter article Glasner and Weiss established a stronger result (for compact metric systems) which implies that a transitive non-minimal compact metric system (X,f) with dense set of almost periodic points is sensitive. This is true also for group actions as was proved in the book of Glasner (Ergodic Theory via Joinings, 2003). Our aim is to generalize these results in the frame of a unified approach for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces.

KW - Almost equicontinuous

KW - Almost periodic point

KW - Semigroup action

KW - Sensitivity

KW - Transitive point

UR - http://www.scopus.com/inward/record.url?scp=37749045838&partnerID=8YFLogxK

U2 - 10.1007/s00233-007-9033-5

DO - 10.1007/s00233-007-9033-5

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AN - SCOPUS:37749045838

SN - 0037-1912

VL - 76

SP - 133

EP - 141

JO - Semigroup Forum

JF - Semigroup Forum

IS - 1

ER -