Large entropy measures for endomorphisms of CP(k)

Let $f$ be an holomorphic endomorphism of $\mathbb{C}\mathbb{P}^k$. We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to $\log d^k$. We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of $f$. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.Comment: 24 page

ONELOUDR AND COMMUNITIES A CONCEPT FOR SHARING AND COLLABORATING ONLINE

This paper explores the evolution of sounds recording as a developed technology through the ages, from the beginning of the twentieth century to the twenty-first century. It highlights the contributions of musicians and researchers alike from the invention of the Théâtrophone by Ader to the improvement of sounds recording by Stokowski and the first recording with a computer by Hiller and Isaacson. This paper also underlines the problem of copyright, which came with the new technologies, and the possibilities to record a radio station on a tape for example. The arrival of Internet as a medium of communication and the new way to share music through the invention of the MP3 format online. The paper also focuses on my collaboration with Oneloudr, which is an online platform for musicians, that lead them to create, share and sell their music