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سایهنگار حد متوسط و تجزیۀ مغلوب
Limit Average Shadowing and Dominated Splitting
جبر
alg
مقاله مستقل
Original Manuscript
<span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">در این مقاله</span></span></span> <span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> ابتدا</span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> مفهوم خاصیت سایهنگار حد متوسط برای وابرسانیهایبر خمینۀ فشردۀ هموا</span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">ر </span></span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" height="18" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="14" ></span></span></span> <span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">معرفی شده و سپس یک رده از وابرسانیهای دارای ویژگی سایهنگار حد متوسط که خاصیت سایهنگار<sup> </sup>ندارند، ارایه شده است. افزون بر آن، ثابت میکنیم که برای مجموعه بستۀ </span></span></span><em><span dir="LTR"><span style="color:black;"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">f</span></span></span></span></em><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">-ناوردا</span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">ی </span></span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" height="18" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="9" ></span></span></span> <span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">از وابرسانی </span></span></span><span dir="LTR"><span style="color:black;"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">f</span></span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> ، اگر </span></span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" height="18" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="9" ></span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> سایهنگار حد متوسط </span></span></span><span dir="LTR"><span style="color:black;"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">C<sup>1</sup></span></span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">-پایدار و نقطههای کمینه </span></span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" height="18" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="9" ></span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> چگال باشد، آنگاه </span></span></span><span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" height="18" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="9" ></span></span></span><span style="color:black;"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> یک تجزیۀ مغلوب میپذیرد.</span></span></span><span dir="LTR"><span style="color:black;"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span></span><a href="./files/site1/files/62/2Tajbakhsh.pdf">./files/site1/files/62/2Tajbakhsh.pdf</a>
<strong>Introduction</strong><br>
The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the average shadowing property. K. Sakai proved that the <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms.<br>
Asymptotic average shadowing (AASP) defined by R. Gu for continuous maps, combines to the limit shadowing property with the average shadowing property. Here we modify the notion (AASP) and define the limit average shadowing for diffeomorphisms (LASP). R. Gu presented some basic properties of the limit average shadowing for continuous maps. He proved that if a continuous map has the limit average shadowing on a compact metric space<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="13" >, then <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="13" >is chain transitive and that <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="8" >-hyperbolic homeomorphisms with limit average shadowing are topologically transitive. M. Kulczycki <em>et.<a name="_GoBack"></a>al.</em>, found some relations between LASP and the other notion of topological dynamics. They proved that a surjective map with specification property has the LASP. Also, they found the relation between LASP and shadowing property. They also have been shown that an expansive continuous map with shadowing property is LASP if and only if it is mixing. This paper follows the ideas of R. Gu and M. Kulczycki <em>et.al.</em> Here we define LASP for diffeomorphism with a slight modification of the continuous case. We give an example which shows that shadowing property and LASP are not equivalent. Also, we introduce the notion of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >- stably limit average shadowing for a closed <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" >-invariant subset <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="14" >of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="16" > and show that if<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="14" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-stably limit average shadowing and the minimal points of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > are dense there, then <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > admits a dominated splitting.<br>
<strong>Statement of the results</strong><br>
In this paper we give a system which has the limit average shadowing, but not the shadowing property. Also, one can give examples which have the shadowing but not the limit average shadowing property. Thus the limit average shadowing property does not imply the shadowing property. In fact, we can give a class of diffeomorphisms which have LASP, but not the shadowing property. In fact the following proposition gives a large class of diffeomorphisms satisfying the limit average shadowing.<br>
<strong>Proposition A. </strong><em>Let </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" ><em> be a locally maximal </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" ><em>-invariant set. If </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" ><em> is the specific set for </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="12" ><em>then </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" ><em> is limit average shadowable.</em><br>
The main purpose of the paper is to characterize the closed <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" >-invariant set via limit average shadowing property in <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-open condition. So, we consider the notion of limit average shadowing property in geometric differential dynamical systems. First we show that if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" > has the limit average shadowing property on a closed <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" >-invariant set <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > then <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > is chain transitive. By using chain transitivity and limit average shadowing property we can prove that <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > is transitive.<br>
<strong>Proposition B. </strong>If <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > is <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-stably limit average shadowable, then there is a neighborhood <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="31" > of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" > and a neighborhood <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="10" > of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > such that <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image026.gif" width="18" >contains neither almost sinks nor almost sources for any <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image028.gif" width="64" ><br>
Since we have proved that if <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image020.gif" width="12" > has the limit average shadowing property on a closed <em>f</em>-invariant set <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > and minimal points of <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="8" > are dense then <img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > is transitive. It is essentially proved that under assumptions and conclusions of the Proposition B,<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="14" > admits a dominated splitting. Thus we get the main result of this paper.<br>
<strong>Theorem C. </strong><em>Let </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" ><em> be a closed f-invariant set whose minimal points are dense there. If </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" ><em> is </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" ><em>-stably limit average shadowing then </em><img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="11" > <em>admits a dominated splitting.<a href="./files/site1/files/62/2Abstract.pdf">./files/site1/files/62/2Abstract.pdf</a></em><br>
<strong> </strong>
سایه نگار : سایه نگار حد متوسط: وابرسانیها: هذلولوی:تجزیه مغلوب
Dominated splitting, Average shadowing, Limit shadowing, Asymptotic average shadowing, specification, Chain transitive, transitive, Mixing.
157
168
http://mmr.khu.ac.ir/browse.php?a_code=A-10-455-1&slc_lang=fa&sid=1
Khosro
Tajbakhsh
خسرو
تاجبخش
khtajbakhsh@modares.ac.ir
10031947532846003744
10031947532846003744
Yes
Tarbiat Modares University
دانشگاه تربیت مدرس، دانشکدۀ علوم ریاضی، گروه ریاضی محض