國立虎尾科技大學 |

Real-Analytic ABC Method on the Torus.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Real-Analytic ABC Method on the Torus./
Author:	Banerjee, Shilpak.
Description:	1 online resource (104 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
Contained By:	Dissertation Abstracts International79-04B(E).
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355329698

Real-Analytic ABC Method on the Torus.
Banerjee, Shilpak.

Real-Analytic ABC Method on the Torus. - 1 online resource (104 pages)

Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.

Thesis (Ph.D.)

Includes bibliographical references

Nearly half a century ago, Anosov and Katok invented a scheme to produce examples of volume preserving smooth zero entropy diffeomorphisms satisfying interesting dynamical properties. Their recipe for creating diffeomorphisms later came to be known as the 'approximation by conjugation' method or the 'Anosov-Katok' method or simply the 'AbC' method. Over the years the AbC method along with its modifications has become a powerful tool for building diffeomorphisms with rich dynamical behavior in the zero entropy set up.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355329698Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

Real-Analytic ABC Method on the Torus.
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100 1 $a Banerjee, Shilpak. $3 1180972
245 1 0 $a Real-Analytic ABC Method on the Torus.
264 0 $c 2017
300 $a 1 online resource (104 pages)
336 $a text $b txt $2 rdacontent
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500 $a Source: Dissertation Abstracts International, Volume: 79-04(E), Section: B.
500 $a Adviser: Anatole Katok.
502 $a Thesis (Ph.D.) $c The Pennsylvania State University $d 2017.
504 $a Includes bibliographical references
520 $a Nearly half a century ago, Anosov and Katok invented a scheme to produce examples of volume preserving smooth zero entropy diffeomorphisms satisfying interesting dynamical properties. Their recipe for creating diffeomorphisms later came to be known as the 'approximation by conjugation' method or the 'Anosov-Katok' method or simply the 'AbC' method. Over the years the AbC method along with its modifications has become a powerful tool for building diffeomorphisms with rich dynamical behavior in the zero entropy set up.
520 $a These diffeomorphisms are obtained as limits of volume preserving periodic diffeomorphisms on compact smooth manifolds admitting a non trivial action of the circle. The periodic diffeomorphisms are constructed to satisfy some finite version of the targeted dynamical property.
520 $a Unfortunately, when one moves from the world of smooth diffeomorphism to the world of real-analytic diffeomorphisms, the story becomes much more complicated. In fact, the problem is largely intractable for abstract real-analytic manifolds, but there is some hope after restricting attention to specific manifolds like a torus or an odd dimensional sphere.
520 $a In this dissertation, we introduce a way to implement the AbC method on a torus and build real-analytic diffeomorphisms. The key idea behind our construction is the fact that step functions can be approximated by real-analytic functions. On the other hand one can build transformations on the torus using these step functions that produce a sliding motion, and we are able to show that a finite composition of such transformations give us enough flexibility to realize a variety of finite versions of dynamical properties on a torus. As application of our implementation, we produce real-analytic versions of several smooth AbC diffeomorphisms.
520 $a For example, we obtain non standard real-analytic realizations of certain irrational rotations of the circle on a torus. Non-standard real-analytic realizations of certain ergodic translations of a torus are also obtained on another torus.
520 $a We note that the work of Fathi and Herman modified the original AbC method and made it possible to control all orbits. Such modified versions known as topological AbC methods made it possible to obtain properties like unique ergodicity and minimality, and we are able to reproduce some of those methods in our set up. In particular we are able to show that non-standard real-analytic realizations of certain irrational rotations of the circle can be obtained on the two dimensional torus as uniquely ergodic diffeomorphisms. We also construct examples of real-analytic minimal diffeomorphisms with a finite number of ergodic invariant measures, each of which are absolutely continuous with respect to the Lebesgue measure.
520 $a Finally, we provide real-analytic realizations of a large class of AbC transformations built by the abstract AbC scheme on a two dimensional torus. These AbC diffeomorphisms we realize are isomorphic to the (strongly uniform) circular systems with fast growing parameters invented by Foreman and Weiss.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a The Pennsylvania State University. $3 845556
773 0 $t Dissertation Abstracts International $g 79-04B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10666427 $z click for full text (PQDT)