Typical dynamics of plane rational maps with equal degrees

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle and one of repelling type. Neither measure is supported in an algebraic curve, and $T\circ f$ is `fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation. Absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map.Comment: Many small changes in accord with referee comments and suggestion

Quantum Dynamical Phase Transition in a Spin-Orbit Coupled Bose Condensate

Spin-orbit coupled bosons can exhibit rich equilibrium phases at low temperature and in the presence of particle-particle interactions. In the case with a 1D synthetic spin-orbit interaction, it has been observed that the ground state of a Bose gas can be a normal phase, stripe phase, or magnetized phase in different experimentally controllable parameter regimes. The magnetized states are doubly degenerate and consist of a many-particle two-state system. In this work, we investigate the nonequilibrium quantum dynamics by switching on an external perturbation to induce resonant couplings between the magnetized phases, and predict the novel quantum spin dynamics which cannot be obtained in the single-particle systems. In particular, due to particle-particle interactions, the transition of the Bose condensate from one magnetized phase to the other is forbidden when the strength of external perturbation is less than a critical value, and a full transition can occur only when the perturbation exceeds such critical strength. This phenomenon manifests itself a quantum dynamical phase transition, with the critical point behavior being exactly solvable. From the numerical simulations and exact analytic studies we show that the predicted many-body effects can be well observed with the current experiments.Comment: 9 pages, 4 figures, plus supplementary materia

Non-Institutional Market Making Behavior: The Dalian Futures Exchange

This paper contains three useful contributions: (1) it collects a new data-set of electronic transaction data on soybean futures from the Dalian Futures Exchange in China that records, not only the usual elements of each transaction (such as price and size) but also identifies broker and customer identities, variables not usually obtainable; (2) it presents new econometric methods for the analysis of dynamic multivariate count data based on the autoregressive conditional intensity model of Jordà and Marcellino (2000); and (3) together, the new data and econometric methods allow us to investigate, in a manner not available before, the determinants and effects of non-institutional market making (or scalping).market making, autoregressive conditional intensity, high-frequency data