Key Address on Dynaculture: Progress as a Deadly Contest

My distinguished colleagues have discussed for us t his morning several vast and vital problem areas faced by this state and this nation today. In what follows, I shall offer you a viewpoint of the current world contest and of the factors which will govern its outcome. I hope thus to provides useful background for our further discussion of these state and national problems

Key Address on Dynaculture: Progress as a Deadly Contest

My distinguished colleagues have discussed for us t his morning several vast and vital problem areas faced by this state and this nation today. In what follows, I shall offer you a viewpoint of the current world contest and of the factors which will govern its outcome. I hope thus to provides useful background for our further discussion of these state and national problems

Dynamical properties of a dissipative discontinuous map: A scaling investigation

The effects of dissipation on the scaling properties of nonlinear discontinuous maps are investigated by analyzing the behavior of the average squared action \left as a function of the $n$-th iteration of the map as well as the parameters $K$ and $\gamma$, controlling nonlinearity and dissipation, respectively. We concentrate our efforts to study the case where the nonlinearity is large; i.e., $K\gg 1$. In this regime and for large initial action $I_0\gg K$, we prove that dissipation produces an exponential decay for the average action \left. Also, for $I_0\cong 0$, we describe the behavior of \left using a scaling function and analytically obtain critical exponents which are used to overlap different curves of \left onto an universal plot. We complete our study with the analysis of the scaling properties of the deviation around the average action $\omega$.Comment: 20 pages, 7 figure

Invariant measures for Cherry flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.Comment: 12 pages; updated versio