Hausdorff dimension of the exceptional set of interval piecewise affine contractions

Let $I=[0,1)$, $-1<\lambda<1$ and $f\colon I\to I$ be a piecewise $\lambda$-affine map of the interval $I$, i.e., there exist a partition $0=a_0<a_1<\cdots< a_{k-1}<a_k=1$ of the interval $I$ into $k\geq2$ subintervals and $b_1,\ldots, b_k\in\mathbb{R}$ such that $f(x)=\lambda x+ b_i$ for every $x\in[a_{i-1},a_{i})$ and $i=1,\ldots,k$. The exceptional set $\mathcal{E}_f$ of $f$ is the set of parameters $\delta\in\mathbb{R}$ such that $R_\delta\circ f$ is not asymptotically periodic, where $R_\delta\colon I\to I$ is the rotation of angle $\delta$. In this paper we prove that $\mathcal{E}_f$ has zero Hausdorff dimension. We derive this result from a more general theorem concerning piecewise Lipschitz contractions on $\mathbb{R}$ that has independent interest.Comment: 12 pages, 2 figure

Ergodicity of polygonal slap maps

Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with non-specular reflections laws. We study the absolutely continuous invariant probabilities of the slap maps for several polygons, including regular polygons and triangles. We also present a general method for constructing polygons with slap maps having more than one ergodic absolutely continuous invariant probability.Comment: 17 pages, 6 figure

EQUINE CONCEPTUS DEVELOPMENT – A MINI REVIEW

Many aspects of early embryonic development in the horse are unusual or unique; this is of scientific interest and, in some cases, considerable practical significance. During early development the number of different cell types increases rapidly and the organization of these increasingly differentiated cells becomes increasingly intricate as a result of various inter-related processes that occur step-wise or simultaneously in different parts of the conceptus (i.e., the embryo proper and its associated membranes and fluid).  Equine conceptus development is of practical interest for many reasons. Most significantly, following a high rate of successful fertilization (71-96%) (Ball, 1988), as many as 30-40% of developing embryos fail to survive beyond the first two weeks of gestation (Ball, 1988), the time at which gastrulation begins. Indeed, despite considerable progress in the development of treatments for common causes of sub-fertility and of assisted reproductive techniques to enhance reproductive efficiency, the need to monitor and rebreed mares that lose a pregnancy or the failure to produce a foal, remain sources of considerable economic loss to the equine breeding industry. Of course, the potential causes of early embryonic death are numerous and varied (e.g. persistent mating induced endometritis, endometrial gland insufficiency, cervical incompetence, corpus luteum (CL) failure, chromosomal, genetic and other unknown factors (LeBlanc, 2004). However, the problem is especially acute in aged mares with a history of poor fertility in which the incidence of embryonic loss between days 2 and 14 after ovulation has been reported to reach 62-73%, and in which embryonic death is due primarily to embryonic defects rather than to uterine pathology (Ball et al., 1989; Carnevale &amp; Ginther, 1995; Ball, 2000).

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the reflected angle towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic SRB measures whose basins cover a set of full Lebesgue measure.Comment: 26 pages, 2 figure

Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation

Consider an analytic two-degrees of freedom Hamiltonian system with an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation, i.e., the eigenvalues of the linearized system at the equilibrium change from complex ±β ±iα (α,β > 0) for ε > 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε < 0. At ε = 0 the equilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign of a certain coefficient of the normal form there are two main bifurcation scenarios. In one of these (the stable case), two dimensional stable and unstable manifolds of the equilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable and unstable manifolds coincide and the invariant manifolds are indistinguishable using classical perturbation theory. In particular, Melnikov’s method is not capable to evaluate the splitting. In this thesis we have addressed the problem of measuring the splitting of these manifolds for small values of the bifurcation parameter ε. We have estimated the size of the splitting which depends on a singular way from the bifurcation parameter. In order to measure the splitting we have introduced an homoclinic invariant ωε which extends the Lazutkin’s homoclinic invariant defined for area-preserving maps. The main result of this thesis is an asymptotic formula for the homoclinic invariant. Assuming reversibility, we have proved that there is a symmetric homoclinic orbit such that its homoclinic invariant can be estimated as follows, ωε = ±2e−πα/2β (ω0 + O(ε1−μ)). where μ > 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic formula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0 then the invariant manifolds intersect transversely. The Stokes constant ω0 is defined for the Hamiltonian at the moment of bifurcation only. We also prove that it does not vanish identically. Finally, we apply our methods to study homoclinic solutions in the Swift-Hohenberg equation. Our results show the existence of multi-pulse homoclinic solutions and a small scale chaos