Resumen
We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

Resumen
The article explores digital watermarking technology from many angles, focusing on the technology and engineering efforts, applications, limitations, attacks, benchmark tests, standardization opportunities, business models, and market acceptance. Digital watermarking technology is making headway in several research areas as well as in commercial products, bringing a recent tide of publicity and controversy. The advent of the Internet has meant new business, scientific, entertainment, and social opportunities in the form of electronic publishing and advertising, real-time information delivery, product ordering, transaction processing, digital repositories and libraries, Web newspapers and magazines, network video and audio, personal communication. Digital watermarking is the embedding of unobtrusive marks or labels that can be represented in bits in digital content. Digital watermarking technology is an emerging field in computer science, cryptography, signal processing, and communications under active research and development