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Smooth TA-maps with Robust Shadowing Are Axiom A

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Seyed Mohsen Moosavi at Tarbiat Modares University
Seyed Mohsen Moosavi
Khosro Tajbakhsh at Tarbiat Modares University
Khosro Tajbakhsh
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Abstract

In this paper we show that smooth TA-endomorphisms of compact manifolds with c-expansivity (that is, expansive in the inverse limit) and C1C^1-stable shadowing are Axiom A.
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https://doi.org/10.1007/s10883-021-09579-6
Smooth TA-maps with Robust Shadowing Are Axiom A
Seyed Mohsen Moosavi1·Khosro Tajbakhsh1
Received: 10 November 2020 / Revised: 1 June 2021 / Accepted: 11 October 2021 /
©The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
Abstract
In this paper, we show that smooth TA-endomorphisms of compact manifolds with
c-expansivity (that is, expansive in the inverse limit) and C1-stable shadowing are Axiom A.
Keywords Axiom A ·Hyperbolicity ·Shadowing ·Endomorphism ·Stable shadowing
Mathematics Subject Classification (2010) Primary: 37D20 ·37C20
1 Introduction
The influence of persistence behavior of a dynamical system on tangent bundle is
always a challenge in dynamical systems. These properties has been studied deeply for
diffeomorphisms. But there are not as many researches done in the non-injective or
local-diffeomorphism (endomorphism) cases. For instance, Ma˜
n´
eR.[9] showed if a diffeo-
morphism fon a compact C∞manifold Mis C1-robustly expansive (roughly speaking,
thereisanopenC1-neighborhood of f,Uf, with all g∈Ufexpansive), on Mthen fis
quasi-Anosov. Also, for diffeomorphisms, focusing on various shadowing properties, such
as usual shadowing, inverse shadowing, limit shadowing, orbital shadowing, some other
interesting results are obtained in [3,6,12,13,16]. An important point of interest about
the non-injective Anosov endomorphisms is that they are structurally unstable. Although
M. Shub had conjectured that by a similar procedure to the expanding maps, non-injective
Anosov endomorphisms are stable in the set of Anosov endomorphisms, Przytyki F. in [14]
showed that they are not so in any case.
Motivated by [3,9,16], the main theorem in this paper is obtained in this direction. A
continuous surjection f:M→M, on a closed topological manifold (i.e. a compact con-
nected topological manifold without boundary), is called a topological Anosov map (abbr.
TA-map) if fis c-expansive and has shadowing property (for definitions, see the next
Khosro Tajbakhsh
khtajbakhsh@modares.ac.ir; arash@cnu.ac.kr
Seyed Mohsen Moosavi
seyedmohsen.moosavi@modares.ac.ir
1Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University,
Tehran 14115-134, Iran
Published online: 13 November 2021
Journal of Dynamical and Control Systems (2023) 29:43–53
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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