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Quantifying the Shadowing Property

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Abstract

In this paper, we present a revised formulation of the Lipschitz shadowing property introduced by Pilyugin and Tikhomirov (Nonlinearity 23:2509–2515, 2010), which involves a numerical quantity known as the “Lipschitz shadowability constant.” This constant is computed for an invertible operator within a Banach space, by taking the reciprocal of the operator’s shadowability constant. We provide a detailed analysis of the properties associated with this constant and calculate its value in the hyperbolic context. Additionally, we offer an estimation for convergent operators as described in Koliha (J Math Anal Appl 48:446–469, 1974). Throughout the paper, we showcase various applications that benefit from this concept.

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Funding

Partially supported by Basic Science Research Program through the NRF funded by the Ministry of Education of the Republic of Korea (Grant Number: 2022R1l1A3053628). CAM was also supported by CNPq-Brazil grant No 307776/2019-0.

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Authors and Affiliations

  1. Department of Mathematics, Chungnam National University, Daejeon, 34134, Republic of Korea

    C. A. Morales & T. Nguyen

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  1. C. A. Morales
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  2. T. Nguyen
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These authors contributed equally to this work.

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Correspondence to C. A. Morales.

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Morales, C.A., Nguyen, T. Quantifying the Shadowing Property. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10317-8

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  • DOI: https://doi.org/10.1007/s10884-023-10317-8

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