Shadowing in the case of nontransverse intersection

Petrov, Aleksey

doi:10.1090/spmj/1378

Cited by 2 publications

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“…Note that the two-dimensional C 0 transversality condition used in the paper [9] has a natural multidimensional generalization (see [10]). At the same time, the multidimensional analog of Theorem 2 does not hold; the paper [11] contains an example of a diffeomorphism of a three-dimensional manifold that satisfies Axiom A, has the Hölder shadowing property, and has two hyperbolic fixed points such that their one-dimensional stable and unstable manifolds have a point of intersection (thus, the C 0 transversality condition formulated in the paper [10] is violated).…”

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Relations between Shadowing and Inverse Shadowing in Dynamical Systems

Petrov

2019

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In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. For some classes of inverse shadowing, we construct examples of homeomorphisms that have the inverse shadowing property but do not have the shadowing property. IntroductionIn the theory of dynamical systems, a lot of publications are devoted to the problems of shadowing of pseudotrajectories and of inverse shadowing (let us mention the monographs [1,2] and also the papers [3,4]).At present, we do not have a complete answer to the question: How are these two properties related?In this paper, we give a review of the existing answers to the above question and extend the list of examples which show that the two properties are nonequivalent.Let us formulate the main definitions which we need. Let f : M → M be a homeomorphism of a metric space (M, dist).Definition 1. Fix a d > 0. We say that a sequence of pointsDefinition 2. We say that f has the shadowing property if for any ε > 0 there exists a d = d(ε) > 0 such that for any d-pseudotrajectory ξ = {ξ i } one can find a point p ∈ M such that the inequalitieshold for any i ∈ Z. In this case, we say that the point p ∈ M ε-shadows the pseudotrajectory ξ.We also need a stronger variant of Definition 2.Definition 3. If there exist constants L, d 0 > 0 and γ ∈ (0, 1) such that for any d-pseudotrajectory ξ of f with d ∈ (0, d 0 ) one can find a point p ∈ M that Ld γ -shadows the pseudotrajectory ξ, we say that the homeomorphism f

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“…In the next section, we show that for a diffeomorphism that satisfies Axiom A, the C 0 transversality condition is necessary for inverse shadowing. Thus, the example constructed in [11] is an example of a diffeomorphism that has the shadowing property and does not have the inverse shadowing property.…”

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Relations between Shadowing and Inverse Shadowing in Dynamical Systems

Petrov

2019

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“…Electrochemistry dates back to the early 19 th century. Pioneering work by Volta [127] Petrov, [128] Faraday, [129] and the Göttingen scholar H. Kolbe [130] revealed the capability of electric current to enable unprecedent reactivities. The oxidative electro-decarboxylation of aliphatic carboxylic acids to form the corresponding dimeric alkanes via radical recombination became the first well-documented organic electrolysis (Scheme 1-27).…”

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Electrochemical Benzylic C−H Fluorination and 4d-Metallaelectro-Catalyzed C−H and C−C Functionalizations

Scheremetjew¹

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but the trend is that the mildest conditions and lowest catalysts loadings can be achieved when expensive palladium catalysts are employed.Scheme 1-2 General mechanism for palladium-catalyzed cross-coupling reactions.These methods nowadays belong to the standard repertoire of organic synthesis and their importance can be hardly overrated, since the production of an enormous number of lifesaving drugs relies on cross-coupling chemistry. These monumental achievements were honored with the Nobel Prize, awarded in 2010 to Akira Suzuki, Ei-ichi Negishi and Richard F. Heck, [20] and the Wolf Prize 2019, awarded to Stephen L. Buchwald and John F. Hartwig. [21] Despite these advances, catalysis in general and coupling reactions in particular remain an attractive research area due to multiple challenges that are still associated with the atom-and step-economy of the envisioned transformations. C−H ActivationThe prefunctionalization of substrates tworards the nucleophile/electrophile pair, that is needed for traditional cross-coupling, causes chemical waste during each synthetic step (Scheme 1-3, a). Moreover, many organometallic reagents call for special handling precautions due to their toxicity and/or hydrolytic sensitivity. An arguably more direct approach is reductive cross-coupling of two electrophiles (Scheme 1-3, b). [22] However, in addition to the innate challenge of overcoming undesired homocoupling, the waste balance is similarly problematic. In this context, C−H activation emerged as a powerful approach to unlock expedient reactivities of ubiquitously occurring C−H bonds while circumventing

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Shadowing in the case of nontransverse intersection

Cited by 2 publications

References 6 publications

Relations between Shadowing and Inverse Shadowing in Dynamical Systems

Relations between Shadowing and Inverse Shadowing in Dynamical Systems

Electrochemical Benzylic C−H Fluorination and 4d-Metallaelectro-Catalyzed C−H and C−C Functionalizations

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