arXiv subjects

Sogo Murakami

Publications and source records attributed to Sogo Murakami.

4 recordsLinked to original sources

arXiv2025

A shadowable chain recurrent set with an attached hyperbolic singularity

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a $C^\infty$-flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto, López, Rego and Sánchez (Math. Annalen 390:417-437).

math.DS↗

arXiv2024

Multidimensional $C^0$ transversality and the shadowing property for Axiom A diffeomorphisms

Petrov and Pilyugin (2015) generalized a notion of $C^0$ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from ${\mathbb R}^n$ to a manifold, which is a purely topological one. They also provided a sufficient condition for the $C^0$ transversality in terms of homological nature. In this paper, we prove that such a homological condition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property. Moreover, it is proved that the $C^0$ transversality of Axiom A diffeomorphisms with codimension one basic sets implies the homological condition.

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arXiv2023

Oriented and standard shadowing properties on closed surfaces

We prove that oriented and standard shadowing properties are equivalent for topological flows on closed surfaces with the nonwandering set consisting of the finite number of critical elements (i.e., singularities or closed orbits). Moreover, we prove that each isolated singularity of a topological flow on a closed surface with the oriented shadowing property is either asymptotically stable, backward asymptotically stable, or admits a neighborhood which splits into two or four hyperbolic sectors.

math.DS↗

arXiv2022

Oriented and standard shadowing properties for topological flows

We prove that oriented and standard shadowing properties are equivalent for topological flows with finite singularites that are Lyapunov stable or Lyapunov unstable. Moreover, we prove that the direct product $ϕ_1 \times ϕ_2$ of two topological flows has the oriented shdowing property if $ϕ_1$ with finite singuralities has the oriented shadowing property, while $ϕ_2$ has the limit set consisting of finite singularities that are Lyapunov stable or Lyapunov unstable.

math.DS↗