Some consequences of shadowing and entropy-expansiveness
Given any continuous self-map of a compact metric space, we prove that a non-empty uniformly rigid subset must be zero-dimensional, under assumptions of shadowing and h-expansiveness.
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arXiv subjects
Noriaki Kawaguchi
Publications and source records attributed to Noriaki Kawaguchi.
On ball expanding maps
We study the dynamical properties of ball expanding maps, a class of continuous self-maps defined on compact metric spaces. For a ball expanding map, we show that: (1) the set of periodic points is dense in the chain recurrent set; (2) if the topological entropy of the map is zero, then the chain recurrent set is finite; (3) the map has only finitely many chain components; (4) if the space is perfect, then the topological entropy of the map is positive; and (5) if the space is connected, then the map is locally eventually onto and hence mixing. Several examples are also provided.
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A note on Bohr chaos and hyperbolic sets
This paper studies the relationship between shadowing phenomena and Bohr chaos in dynamical systems. We provide sufficient conditions for Bohr chaos in terms of shadowing. By combining those conditions with the shadowing lemma, we obtain some results on Bohr chaos and hyperbolic sets. Our results also highlight some simple but non-trivial structural properties of hyperbolic sets.
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A note on the omega-chaos
For any continuous self-map of a compact metric space, we provide sufficient conditions under which the infinite direct product of the map is $ω$-chaotic. We also apply the result to obtain some examples of unusual $ω$-chaotic maps.
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On non-chaotic hyperbolic sets
We give necessary and sufficient conditions for a hyperbolic set to be non-chaotic (or, conversely, chaotic) in a certain sense.
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Shadowing and the continuity of omega-limit sets
This paper examines the relationship between shadowing phenomena and the continuity properties of $ω$-limit sets in dynamical systems. We give a necessary and sufficient condition for a shadowable point to be an upper (resp. a lower) semicontinuity point of $ω$-limit sets. Assuming global shadowing, we show that the lower semicontinuity of $ω$-limit sets is equivalent to the chain continuity. We also show that the lower semicontinuity of $ω$-limit sets is equivalent to the chain continuity in a general setting. Several examples are given to illustrate the results.
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Shadowing, chain transitive sets with nonempty interior and attractor boundaries
We examine certain phenomena in $C^1$-dynamics from a viewpoint of shadowing and improve a known result on hyperbolic sets. We also review a result on the stability of attractor boundaries from the same viewpoint and derive several additional results.
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Shadowing, sensitivity and entropy points
For continuous self-maps of compact metric spaces, we explore the relationship among the shadowable points, sensitive points, and entropy points. Specifically, we show that (1) if the set of shadowable points is dense in the phase space, then a point located in the interior of the set of sensitive points is an entropy point; and (2) if the topological entropy is zero, then the denseness of the set of shadowable points is equivalent to almost chain continuity. In addition, we present a counter-example to a question raised by Ye and Zhang regarding entropy points.
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Sufficient conditions for distributional chaos of type I
Distributional chaos of type I (DC1) is a stronger variant of Li-Yorke chaos. In this paper, we consider the fact that the time-one map of a mixing Anosov flow exhibits DC1 and generalize it to obtain simple sufficient conditions for DC1.
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A note on odometers and shadowing
For a continuous self-map of a compact metric space, we provide a sufficient condition for the orbit of a point to converge to a periodic orbit or an odometer. We show that if a continuous self-map of a compact metric space has the shadowing property, then the set of points whose orbits converge to periodic orbits or odometers is dense.
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Some results on s-limit shadowing and Li-Yorke type chaos
In [13], for any continuous self-map of a compact metric space with s-limit shadowing, by using a $G_δ$-partition of the phase space, a global description of Li-Yorke type chaos (with respect to several particular Furstenberg families) is obtained. In this paper, we complement the results in [13] by some results concerning the s-limit shadowing and general Furstenberg families. An example is also given to illustrate the results.
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Shadowing and the basins of terminal chain components
We provide an alternative view of some results in [1, 3, 11]. In particular, we prove that (1) if a continuous self-map of a compact metric space has the shadowing, then the union of the basins of terminal chain components is a dense $G_δ$-subset of the space; and (2) if a continuous self-map of a locally connected compact metric space has the shadowing, and if the chain recurrent set is totally disconnected, then the map is almost chain continuous.
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Contractive shadowing property of dynamical systems
For topological dynamical systems defined by continuous self-maps of compact metric spaces, we consider the contractive shadowing property, i.e., the Lipschitz shadowing property such that the Lipschitz constant is less than 1. We prove some basic properties of contractive shadowing and show that non-degenerate homeomorphisms do not have the contractive shadowing property. Then, we consider the case of ultrametric spaces. We also discuss the spectral decomposition of the chain recurrent set under the assumption of contractive shadowing. Several examples are given to illustrate the results.
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Some results on shadowing and local entropy properties of dynamical systems
We consider some local entropy properties of dynamical systems under the assumption of shadowing. In the first part, we give necessary and sufficient conditions for shadowable points to be certain entropy points. In the second part, we give some necessary and sufficient conditions for (non) h-expansiveness under the assumption of shadowing and chain transitivity; and use the result to present a counter-example for a question raised by Artigue et al. [Proc. Amer. Math. Soc. 150 (2022), 3369-3378].
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S-limit shadowing and a global description of Li-Yorke type chaos
For any continuous self-map of a compact metric space, we extend a partition of each chain component with respect to a chain proximal relation to a $G_δ$-partition of the phase space. Under the assumption of s-limit shadowing, we use this partition to give a global description of Li-Yorke type chaos corresponding to several Furstenberg families.
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S-limit shadowing implies the denseness of chain components with the shadowing property
For any continuous self-map of a compact metric space, we consider the space of chain components and prove that the s-limit shadowing implies the denseness of chain components with the shadowing property. It gives a partial answer to a question raised by Moothathu [Topology Appl. 158 (2011) 2232--2239]. We also prove that the s-limit shadowing is not generic in the space of homeomorphisms of a closed differentiable manifold.
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Shadowing, transitivity and a variation of omega-chaos
We study a special type of shadowing (DSP) of chain transitive continuous self-maps of compact Hausdorff spaces. We prove some basic properties of DSP. As application of DSP, we obtain sufficient conditions for a statistical variant of $ω$-chaos and prove the topological genericity of it. We also consider topological distribution of irregular points under the assumption of DSP.
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Generic and dense distributional chaos with shadowing
For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some equivalent properties to generic uniform (distributional) chaos in terms of chains and prove their equivalence to generic (distributional) chaos under the shadowing. Several examples illustrating the main results are also given.
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