Keonhee Lee

A simple method for the compensation of the nonlinearity in the heterodyne interferometer

The accuracy of a heterodyne interferometer is often limited by the periodic nonlinearity that arises mainly from imperfect separation of the two optical frequencies. This paper describes a new method for compensation of the nonlinearity in the heterodyne laser interferometer using the quadrature mixing technique for the phase measurement. The compensation technique is based on the elliptical fitting of two phase-quadrature signals obtained by a lock-in amplifier. The brief analysis and compensation scheme of the nonlinearity in the heterodyne interferometer, and the experimental result using a Zeeman stabilized He-Ne laser, have been presented. The results show that the suggested method can compensate for the nonlinearity of the heterodyne interferometer with sub-nanometre accuracy.

Continuous inverse shadowing and hyperbolicity

We study the concepts of continuous shadowing and continuous inverse shadowing with respect to various classes of admissible pseudo orbits, and characterize hyperbolicity and structural stability using the notion of continuous inverse shadowing.

SHADOWABLE CHAIN TRANSITIVE SETS OF C 1 -GENERIC DIFFEOMORPHISMS

We prove that a locally maximal chain transitive set of a C1-generic diffeomorphism is hyperbolic if and only if it is shadowable.

WEAK INVERSE SHADOWING AND GENERICITY

We study the genericity of the flrst weak inverse shad- owing property and the second weak inverse shadowing property in the space of homeomorphisms on a compact metric space, and show that every shift homeomorphism does not have the flrst weak inverse shadowing property but it has the second weak inverse shadowing property.

Stably inverse shadowable transitive sets and dominated splitting

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we show that if f has the C1-stably inverse shadowing property on a transitive set, then it admits a dominated splitting.

Characterisations of Ω-stability and structural stability via inverse shadowing

We give characterisations of Ω-stable diffeomorphisms and structurally stable diffeomorphisms via the notions of weak inverse shadowing and orbital inverse shadowing, respectively. More precisely, it is proved that the C1 interior of the set of diffeomorphisms with the weak inverse shadowing property coincides with the set of Ω-stable diffeomorphisms and the C1 interior of the set of diffeomorphisms with the orbital inverse shadowing property coincides with the set of structurally stable diffeomorphisms.

Inverse shadowing for structurally stable flows

We introduce the notion of inverse shadowing for continuous flows, and show that a structurally stable flow generated by a C 1 vector field on a compact smooth manifold has the inverse shadowing property with respect to the classes of continuous methods.

Generic diffeomorphisms with measure-expansive homoclinic classes

We prove that for C1 generic diffeomorphisms, every measure-expansive locally maximal homoclinic class is hyperbolic.

INVERSE SHADOWING FOR EXPANSIVE FLOWS

We extend the notion of inverse shadowing defined for dieomorphisms to flows, and show that an expansive flow on a compact manifold with the shadowing property has the inverse shadowing property with respect to the classes of continuous meth- ods. As a corollary we obtain that a hyperbolic flow also has the inverse shadowing property with respect to the classes of continuous methods.

Homoclinic classes with shadowing

We show that for C1 generic diffeomorphisms, an isolated homoclinic class is shadow-able if and only if it is a hyperbolic basic set.Mathematics Subject Classification 2000: 37C20; 37C05; 37C29; 37D05.