Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures
Round 1
Reviewer 1 Report
no comments
Comments for author File: 
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Author Response
Thank you very much for the good rating to our paper, we are really so glad to receive positive reports. In this paper, we introduced the notions of shadowable Lebesgue measure to describe the dynamics of example on T^2 from the measure theoretical view point. In general, it is not so easy to describe the dynamics of a dynamical system by making use of Lebesgue measures in the context of C^1 topology (usually, we need C^2 assumption). In this paper, we have succeeded in describing the dynamics using Lebesgue measures in the context of C^1 topology. Thank you very much. With My Best Regards.
Reviewer 2 Report
The content of the paper is quite unusual for a journal as Axioms. Please reformulate the content. Every theorem must to have assigned his own proof, in a section named as Main results, not separately in a section. Moreover, the paper have no applications section. In this form it is mandatory to reject it.
Author Response
Dear Reviewer, Thank you very much for your comment. With My Best Regards.
Reviewer 3 Report
The paper is very interesting. Pseudo-orbit shadowing property is very important in the study of differentiable dynamical systems. Recently, some authors investigate the shadowable measure, which is a generalization of the shadowing property from the measure theoretical viewpoint. The authors generalized these notions and they introduce the notion of filtrated pseudo-orbit shadowing property and approximately shadowable Lebesgue measure.
In this paper, the authors tried to study filtrated pseudo-orbit shadowing property. Firstly, they prove that if $f$ having the filtrated pseudo-orbit shadowing property, then $f$ admits an approximately shadowable Lebesgue measure. Secondly, they also prove that the $C^1$-interior of the set of diffeomorphisms possessing the filtrated pseudo-orbit shadowing property is characterized as the set of diffeomorphisms satisfying both Axiom A and the no-cycle condition. As a corollary, they prove that if $f$ is a quasi-Anosov diffeomorphism that is not Anosov, then there is a $C^1$-open set (a neighborhood of $f$) such that every element in the set does not have the shadowing property but has an approximately shadowable Lebesgue measure.
The proof is correct and clear. I would like to recommend this paper to be accepted by \emph{Axioms} after they modify their paper by following the suggestion below.
$\bullet$ Page 5, Line 8, $e^{2\pi(\theta+\alpha)}$ should be $e^{2\pi i(\theta+\alpha)}$.
Author Response
Thank you very much for the good rating to our paper, we are really so glad to receive positive reports. In this paper, we introduced the notions of shadowable Lebesgue measure to describe the dynamics of example on T^2 from the measure theoretical view point. In general, it is not so easy to describe the dynamics of a dynamical system by making use of Lebesgue measures in the context of C^1 topology (usually, we need C^2 assumption). In this paper, we have succeeded in describing the dynamics using Lebesgue measures in the context of C^1 topology. Thank you very much. With My Best Regards.
