Measures of Intermediate Entropies for Skew Product Diffeomorphisms

In this paper we study a skew product map $F$ with a measure $\mu$ of positive entropy. We show that if on the fibers the map are $C^{1+\alpha}$ diffeomorphisms with nonzero Lyapunov exponents, then $F$ has ergodic measures of intermediate entropies. To construct these measures we find a set on which the return map is a skew product with horseshoes along fibers. We can control the average return time and show the maximum entropy of these measures can be arbitrarily close to $h_\mu(F)$.Comment: 12 pages, a few mistakes corrected, some sections seriously rewritte

Inflationary electroweak/particle phase transitions and new classical gravitational waves on CMB

Particle phase transitions in the early universe including electroweak and grand unification ones are well-studied subjects. We point out that there are new possible particle phase transitions around inflation. Those new inflationary particle phase transitions, if of the first order, may yield low-frequency gravitational waves (GWs) due to bubble dynamics, leaving imprints on the cosmic microwave background (CMB). In contrast to the nearly scale-invariant primordial GWs caused by vacuum fluctuation, these bubble-generated GWs are classical and have scale dependent B-mode spectra. If decoupled from inflaton, the electroweak phase transition during inflation may serve as a mirror image of the one after reheating where the baryon asymmetry could be generated via electroweak baryogenesis (EWBG). The second new electroweak phase transition may also be the source for EWBG.Comment: 6 pages, 4 figures, prepared for the proceeding of the 2nd LeCosPA Symposium "Everything About Gravity", based on arxiv:1512.0753