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Hyperbolicity, shadowing, and convergent operators

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Abstract

We use the shadowing property in dynamics as reported by Aoki and Hiraide (Topological theory of dynamical systems: recent advances, North-Holland Publishing Co., Amsterdam, 1994) to characterize uniformly convergent operators on Banach spaces (Koliha in J Math Anal Appl 48: 446–469, 1974; Koliha in J Math Anal Appl 43: 778–794, 1973). Next, we examine variations of hyperbolicity for operators as the s-hyperbolicity and generalized s-hyperbolicity. We show that an operator is hyperbolic if and only if it is s-hyperbolic and has the shadowing property. In particular, a s-hyperbolic operator which is neither generalized hyperbolic nor uniformly expansive exists. We finish by locating the homoclinic points of a generalized s-hyperbolic operator. Some applications are given.

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Authors and Affiliations

  1. Department of Mathematics, Chonnam National University, Gwangju, 61186, Republic of Korea

    J. Lee

  2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, Rio de Janeiro, 21945-970, Brazil

    C. A. Morales

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  1. J. Lee
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Correspondence to C. A. Morales.

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Communicated by H. Bruin.

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JL was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A6A3A01091340). CAM was partially supported by Basic Science Research Program through the NRF funded by the Ministry of Education (Grant Number: 2022R1l1A3053628) and CNPq-Brazil Grant No. 307776/2019-0.

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Lee, J., Morales, C.A. Hyperbolicity, shadowing, and convergent operators. Monatsh Math 202, 541–554 (2023). https://doi.org/10.1007/s00605-023-01871-w

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