Abstract
Galileo, in the seventeenth century, observed that the small oscillations of a pendulum seem to have constant period. In fact, the Taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equilibrium. It is well known that, for small oscillations of the pendulum and small intervals of time, the dynamics of the pendulum can be approximated by the dynamics of the harmonic oscillator. We study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected. We analyze such dynamical behaviour through a renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence, we obtain a universal asymptotic focal decomposition for this family of mechanical systems. This paper is intended to be the first in a series of articles aiming at a semiclassical quantization of systems of the pendulum type as a natural application of the focal decomposition associated to the two-point boundary value problem.

Abstract

Abstract
We show a one-to-one correspondence between circle diffeomorphism sequences that are C1+ n-periodic points of renormalization and smooth Markov sequences. © Springer Science+Business Media New York 2014.

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Abstract
We introduce an extension to Merton's famous continuous time model of optimal consumption and investment, in the spirit of previous works by Pliska and Ye, to allow for a wage earner to have a random lifetime and to use a portion of the income to purchase life insurance in order to provide for his estate, while investing his savings in a financial market comprised of one risk-free security and an arbitrary number of risky securities driven by multi-dimensional Brownian motion. We then provide a detailed analysis of the optimal consumption, investment and insurance purchase strategies for the wage earner whose goal is to maximize the expected utility obtained from his family consumption, from the size of the estate in the event of premature death, and from the size of the estate at the time of retirement. We use dynamic programming methods to obtain explicit solutions for the case of discounted constant relative risk aversion utility functions and describe new analytical results which are presented together with the corresponding economic interpretations.

Abstract
Let f and g be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy h between f and g is C-1 at a point in the nearby expanding set of f, then h is a smooth diffeomorphism in the basin of attraction of a renormalization interval of f. In particular, if f:I -> I and g : J -> J are C-r unimodal maps and h is C-1 at a boundary of I, then h is C-r in I.