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# On Almost Shadowable Measures

Received 25 March 2022; accepted 21 April 2022

2022, Vol. 18, no. 2, pp.  297-307

Author(s): Lee K., Rojas A.

In this paper we study the almost shadowable measures for homeomorphisms on compact metric spaces. First, we give examples of measures that are not shadowable. Next, we show that almost shadowable measures are weakly shadowable, namely, that there are Borelians with a measure close to 1 such that every pseudo-orbit through it can be shadowed. Afterwards, the set of weakly shadowable measures is shown to be an $F_{\sigma \delta}$ subset of the space of Borel probability measures. Also, we show that the weakly shadowable measures can be weakly* approximated by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure with respect to any weakly shadowable measure. We show that the notions of shadowableness, almost shadowableness and weak shadowableness coincide for finitely supported measures, or, for every measure when the set of shadowable points is closed. We investigate the stability of weakly shadowable expansive measures for homeomorphisms on compact metric spaces.
Keywords: Shadowable measures, Shadowable points, Homeomorphism
Citation: Lee K., Rojas A., On Almost Shadowable Measures, Rus. J. Nonlin. Dyn., 2022, Vol. 18, no. 2, pp.  297-307
DOI:10.20537/nd220210

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