"Uniform Measures On Inverse Limit Spaces"

Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let f be a function from a compact metric space X into itself where f is continuous, onto and piecewise one-to-one. Let Y be the inverse limit of (X,f). Then starting with a measure m1 on the Borel sets of X, we recursively construct a sequence of probability measures (m1,m2,...) on the Borel sets of X satisfying mn(A)=mn+1[B] for each Borel set A and n=1,2,... and B is the preimage of A under f. This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y. If m1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define an SRB measure or a metric attractor for the shift map on the inverse limit space.Inverse Limits, probability measure, multiple equilibria, global indeterminancy

Chaos and Sector-specific Externalities

Benhabib and Farmer (1996) explore the possibility of local indeterminacy in a twosector model with sector-speci c externalities. They nd that very small sector-specific externalities are su cient for local indeterminacy. In this case, it is possible to construct sunspot equilibria where extrinsic uncertainty matters. In this paper, I provide a global analysis of their model revealing the existence of Euler equation branching. This branching allows for regime switching equilibria with cycles and chaotic behavior. These equilibria occur whether the \local dynamics" are determinate or indeterminate.two-sector model, regime switching, global indeterminacy, cycles and chaos

"Euler Equation Branching"

Some macroeconomic models exhibit a type of global indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion. In this paper, we show that in models with Euler equation branching there are multiple equilibria and that the dynamics are chaotic. In particular, we provide sufficient conditions for a dynamical system on the plane with Euler equation branching to be chaotic and show analytically that in a neighborhood of a steady state, these sufficient conditions will typically be satisfied. We also extend the results of Christiano and Harrison (JME, 1999) for the one-sector growth model with a production externality. In a more general setting, we provide necessary and sufficient conditions for Euler equation branching in this model. We show that chaotic and cyclic equilibria are possible and that this behavior is not dependent on the steady state being "locally" determinate or indeterminate.global indeterminacy, Euler equation branching, multiple equilibria, cycles,chaos, increasing returns to scale, externality, regime switching

Expected Utility in Models with Chaos

In this paper, we provide a framework for calculating expected utility in models with chaotic equilibria and consequently a framework for ranking chaos. Suppose that a dynamic economic model’s equilibria correspond to orbits generated by a chaotic dynamical system f : X ! X where X is a compact metric space and f is continuous. The map f could represent the forward dynamics xt+1 = f(xt) or the backward dynamics xt = f(xt+1). If f represents the forward/backward dynamics, the set of equilibria forms a direct/inverse limit space. We use a natural f-invariant measure on X to induce a measure on the direct/inverse limit space and show that this induced measure is a natural ¾-invariant measure where ¾ is the shift operator. We utilize this framework in the cash-in-advance model of money where f is the backward map to calculate expected utility when equilibria are chaotic.chaos, inverse limits, direct limits, natural invariant measure, cash-in-advance

An approach to optimum subsonic inlet design

Inlet operating requirements are compared with estimated inlet separation characteristics to identify the most critical inlet operating condition. This critical condition is taken to be the design point and is defined by the values of inlet mass flow, free-stream velocity and inlet angle of attack. Optimum flow distributions on the inlet surface were determined to be a high, flat top Mach number distribution on the inlet lip to turn the flow quickly into the inlet and a flat bottom skin-friction distribution on the diffuser wall to diffuse the flow rapidly and efficiently to the velocity required at the fan face. These optimum distributions are then modified to achieve other desirable flow characteristics. Example applications are given

Optimum subsonic, high-angle-of-attack nacelles

The optimum design of nacelles that operate over a wide range of aerodynamic conditions and their inlets is described. For low speed operation the optimum internal surface velocity distributions and skin friction distributions are described for three categories of inlets: those with BLC, and those with blow in door slots and retractable slats. At cruise speed the effect of factors that reduce the nacelle external surface area and the local skin friction is illustrated. These factors are cruise Mach number, inlet throat size, fan-face Mach number, and nacelle contour. The interrelation of these cruise speed factors with the design requirements for good low speed performance is discussed

Theoretical flow characteristics of inlets for tilting-nacelle VTOL aircraft

The results of a theoretical investigation of geometric variables for lift-cruise-fan, tilting nacelle inlets operating at high incidence angles are presented. These geometric variables are investigated for their effects on surface static to free stream pressure ratio, and the separation parameters of maximum to diffuser exit surface velocity ratio and maximum surface Mach number for low speed operating conditions. The geometric parameters varied were the internal lip contraction ratio, external forebody to diffuser exit diameter ratio external forebody length to diameter ratio and internal lip major to minor axis ratio