## Osaka Journal of Mathematics

### Measure-expansive homoclinic classes

#### Abstract

Let $p$ be a hyperbolic periodic point of a diffeomorphism $f$ on a compact $C^{\infty}$ Riemannian manifold $M$. In this paper we introduce the notion of $C^{1}$ stably measure expansiveness of closed $f$-invariant sets, and prove that (i) the chain recurrent set $\mathcal{R}(f)$ of $f$ is $C^{1}$ stably measure expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, and (ii) the homoclinic class $H_{f}(p)$ of $f$ associated to $p$ is $C^{1}$ stably measure expansive if and only if $H_{f}(p)$ is hyperbolic.

#### Article information

Source
Osaka J. Math., Volume 53, Number 4 (2016), 873-887.

Dates
First available in Project Euclid: 4 October 2016

https://projecteuclid.org/euclid.ojm/1475601821

Mathematical Reviews number (MathSciNet)
MR3554846

Zentralblatt MATH identifier
1367.37032

#### Citation

Lee, Keonhee; Lee, Manseob. Measure-expansive homoclinic classes. Osaka J. Math. 53 (2016), no. 4, 873--887. https://projecteuclid.org/euclid.ojm/1475601821

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