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Shadowing and Hyperbolicity 1st ed. 2017 [Pehme köide]

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  • Formaat: Paperback / softback, 218 pages, kõrgus x laius: 235x155 mm, kaal: 3577 g, 5 Illustrations, black and white; XIV, 218 p. 5 illus., 1 Paperback / softback
  • Sari: Lecture Notes in Mathematics 2193
  • Ilmumisaeg: 02-Sep-2017
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319651838
  • ISBN-13: 9783319651835
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Shadowing and Hyperbolicity 1st ed. 2017Suurem pilt
  • Formaat: Paperback / softback, 218 pages, kõrgus x laius: 235x155 mm, kaal: 3577 g, 5 Illustrations, black and white; XIV, 218 p. 5 illus., 1 Paperback / softback
  • Sari: Lecture Notes in Mathematics 2193
  • Ilmumisaeg: 02-Sep-2017
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319651838
  • ISBN-13: 9783319651835
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319651835.html
Märksõnad:
Focusing on the theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems, this book surveys recent progress in establishing relations between shadowing and such basic notions from the classical theory of structural stability as hyperbolicity and transversality.

Special attention is given to the study of "quantitative" shadowing properties, such as Lipschitz shadowing (it is shown that this property is equivalent to structural stability both for diffeomorphisms and smooth flows), and to the passage to robust shadowing (which is also equivalent to structural stability in the case of diffeomorphisms, while the situation becomes more complicated in the case of flows).

Relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets are also described.

The book will allow young researchers in the field of dynamical systems to gain a better understanding of new ideas in the global qualitative theory. It will also be of interest to specialists in dynamical systems and their applications.

Arvustused

The book is clearly written and appropriate both for advanced graduate students in the area and for researchers working or being interested in the field. (Christian Pötzsche, zbMATH 1426.37004, 2020) This book gives an up-to-date account of results on the relations between shadowing and such basic notions from the classical theory of structural stability as hyperbolicity and transversality. The style of presentation is very clear and, in my opinion, the book is quite suitable for researchers in the field of dynamical systems to understand the global qualitative theory from different points of view. (Yujun Zhu, Mathematical Reviews, July, 2018)

1 Main Definitions and Basic Results
1(36)
1.1 Pseudotrajectories and Shadowing in Dynamical Systems with Discrete Time: Chain Transitive Sets
1(8)
1.2 Pseudotrajectories and Shadowing in Dynamical Systems with Continuous Time
9(3)
1.3 Hyperbolicity, Ω-Stability, Structural Stability, Dominated Splittings
12(14)
1.4 Hyperbolic Shadowing
26(11)
2 Lipschitz and Holder Shadowing and Structural Stability
37(88)
2.1 Maizel' and Pliss Theorems
38(13)
2.2 Mane Theorem
51(16)
2.3 Diffeomorphisms with Lipschitz Shadowing
67(8)
2.4 Lipschitz Periodic Shadowing for Diffeomorphisms
75(15)
2.5 Holder Shadowing for Diffeomorphisms
90(12)
2.6 A Homeomorphism with Lipschitz Shadowing and a Nonisolated Fixed Point
102(7)
2.7 Lipschitz Shadowing Implies Structural Stability: The Case of a Vector Field
109(16)
2.7.1 Discrete Lipschitz Shadowing for Flows
110(5)
2.7.2 Rest Points
115(3)
2.7.3 Hyperbolicity of the Chain Recurrent Set
118(1)
2.7.4 Transversality of Stable and Unstable Manifolds
119(6)
3 C1 Interiors of Sets of Systems with Various Shadowing Properties
125(56)
3.1 C1 Interior of SSPD
126(7)
3.2 Diffeomorphisms in Int1 (SSPD) Satisfy Axiom A
133(22)
3.3 Vector Fields in Int1 (OrientSPF \ B)
155(17)
3.4 Vector Fields of the Class B
172(9)
4 Chain Transitive Sets and Shadowing
181(28)
4.1 Examples of Chain Transitive Sets (Homoclinic Classes)
181(6)
4.1.1 Chain Transitive Sets Without Periodic Points
183(1)
4.1.2 Hyperbolic Horseshoes
183(1)
4.1.3 Horseshoe with a Homoclinic Tangency
184(1)
4.1.4 Critical Saddle-Node Horseshoe
185(2)
4.2 C1 - Stably Shadowing Chain Transitive Sets
187(16)
4.2.1 Preliminaries
188(2)
4.2.2 Construction of the Dominated Splitting and Its Extension
190(4)
4.2.3 Proof of Theorem
194(7)
4.2.4 Proof of Corollary
201(2)
4.3 Chain Transitive Sets with Shadowing for Generic Diffeomorphisms
203(6)
References 209(6)
Index 215