Deterministic Dynamics and Chaos: Epistemology and Interdisciplinary Methodology

We analyze, from a theoretical viewpoint, the bidirectional interdisciplinary relation between mathematics and psychology, focused on the mathematical theory of deterministic dynamical systems, and in particular, on the theory of chaos. On one hand, there is the direct classic relation: the application of mathematics to psychology. On the other hand, we propose the converse relation which consists in the formulation of new abstract mathematical problems appearing from processes and structures under research of psychology. The bidirectional multidisciplinary relation from-to pure mathematics, largely holds with the "hard" sciences, typically physics and astronomy. But it is rather new, from the social and human sciences, towards pure mathematics

Pseudo-physical measures for typical continuous maps of the interval

We study the measure theoretic properties of typical C 0 maps of the interval. We prove that any ergodic measure is pseudo-physical, and conversely, any pseudo-physical measure is in the closure of the ergodic measures, as well as in the closure of the atomic measures. We show that the set of pseudo-physical measures is meager in the space of all invariant measures. Finally, we study the entropy function. We construct pseudo-physical measures with infinite entropy. We also prove that, for each m $\ge$ 1, there exists infinitely many pseudo-physical measures with entropy log m, and deduce that the entropy function is neither upper semi-continuous nor lower semi-continuous

Ergodic Measures with Infinite Entropy

We construct ergodic probability measures with infinite imetric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure

Equilibrium States and SRB-like measures of C1 Expanding Maps of the Circle

For any C1 expanding map f of the circle we study the equilibrium states for the potential -log |f'|. We formulate a C1 generalization of Pesin's Entropy Formula that holds for all the SRB measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C1-generic case Pesin's Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non generic case for which no SRB measure exists.Comment: Un this version we include some addings and corrections that were suggested by the referees. The final version will appear in Portugaliae Mathematica and will be available at http://www.ems-ph.org/journals/journal.php?jrn=p