Chaos and a Resonance Mechanism for Structure Formation in Inflationary Models

We exhibit a resonance mechanism of amplification of density perturbations in inflationary mo-dels, using a minimal set of ingredients (an effective cosmological constant, a scalar field minimally coupled to the gravitational field and matter), common to most models in the literature of inflation. This mechanism is based on the structure of homoclinic cylinders, emanating from an unstable periodic orbit in the neighborhood of a saddle-center critical point, present in the phase space of the model. The cylindrical structure induces oscillatory motions of the scales of the universe whenever the orbit visits the neighborhood of the saddle-center, before the universe enters a period of exponential expansion. The oscillations of the scale functions produce, by a resonance mechanism, the amplification of a selected wave number spectrum of density perturbations, and can explain the hierarchy of scales observed in the actual universe. The transversal crossings of the homoclinic cylinders induce chaos in the dynamics of the model, a fact intimately connected to the resonance mechanism occuring immediately before the exit to inflation.Comment: 4 pages. This essay received an Honorable Mention from the Gravity Research Foundation, 1998-Ed. To appear in Mod. Phys. Lett.

A Note on the Shannon Entropy of Short Sequences

For source sequences of length L symbols we proposed to use a more realistic value to the usual benchmark of number of code letters by source letters. Our idea is based on a quantifier of information fluctuation of a source, F(U), which corresponds to the second central moment of the random variable that measures the information content of a source symbol. An alternative interpretation of typical sequences is additionally provided through this approach.Comment: 3 figure

Correction to: Some results on the p(u)-Laplacian problem

Correction to: Mathematische Annalen https://doi.org/10.1007/s00208-019-01803-w In the Original Publication of the article, few errors have been identified in section 5 and acknowledgements section.Agência financiadora Ministry of Education and Science, Russian Federation 117198 Portuguese Foundation for Science and Technology (FCT), Portugal SFRH/BSAB/135242/2017info:eu-repo/semantics/publishedVersio