Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then t...

We prove that a Banach space X has normal structure provided it contains a finite codimensional subspace Y such that all spreading models for Y have normal structure. We show that a Banach space X is strictly convex if the set of fixed points of any nonexpansive map defined in any convex subset C C X is convex and give a sufficient condition for uniform convexity of a space in terms of nonexpan...

We present an analysis of bounded-energy low-tension maps between 2-spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop, we show that we can establish a ‘quantization estimate’ which constrains the energy of the map to lie near to a discrete energy spectrum. One application is to the asymptotics of the harmonic map flow; we find u...

It is reported that the characteristic relations of type-II and type-III intermittencies, with respective local Poincaré maps of xn+1 = (1+ǫ)xn+ax 3 n and xn+1 = −(1 + ǫ)xn − ax3n, are both −ln(ǫ) under the assumption of uniform reinjection probability. However, the intermittencies have various characteristic relations such as ǫ−ν (1/2 ≤ ν ≤ 1) depending on the reinjection probability. In this ...

Conventional maps of election results can give a misleading picture of the popular support that candidates have because population is highly non-uniform and equal areas on a map may not correspond to equal numbers of voters. Taking the example of the 2004 United States presidential election, we show how this problem can be corrected using a cartogram—a map in which the sizes of regions such as ...

Here for a smooth bounded Euclidean domain Ω and a smooth Riemannian manifold N , we show that almost every smooth map φ : ∂Ω→ N serves as boundary data for at most one energy minimizing map u : Ω→ N [Theorem 5.2]. We also obtain some uniform boundary regularity estimates for energy minimizers [Theorem 2.1], which not only are important to our proof, but also imply some other properties of ener...

Facing the competing demands for wider field of view and higher spatial resolution, computer vision will evolve toward greater use of fovea1 sensors and frequent camera movements. Integration of visual information across movements becomes a fundamental problem. We show that integration is possible using a biologically-inspired representation we call the visual-motor calibration map. The map is ...

We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M . Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L1-norm with the map. As a main application we show that the open class of maps introdu...

In this paper we study dynamical properties of the area preserving Hénon map, as a discrete version of open Hamiltonian systems, that can exhibit chaotic scattering. Exploiting its geometric properties we locate the exit and entry sets, i.e. regions through which any forward, respectively backward, unbounded orbit escapes to infinity. In order to get the boundaries of these sets we prove that t...

Let q be a prime and K = Q( √ −D) be an imaginary quadratic field such that q is inert in K. If q is a prime above q in the Hilbert class field of K, there is a reduction map rq : E``(OK) −→ E``(Fq2) from the set of elliptic curves over Q with complex multiplication by the ring of integers OK to the set of supersingular elliptic curves over Fq2 . We prove a uniform asymptotic formula for the nu...