Non-dense orbits on homogeneous spaces and applications to geometry and number theory

Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

Solitons of the midpoint mapping and affine curvature

For a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

On Some Symmetries of Quadratic Systems

We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.

On the vanishing of relative negative K-theory

In this paper, we study the relative negative [Formula: see text]-groups [Formula: see text] of a map [Formula: see text] of schemes. We prove a relative version of Weibel’s conjecture; i.e. if [Formula: see text] is a smooth affine map of Noetherian schemes with [Formula: see text], then [Formula: see text] for [Formula: see text] and the natural map [Formula: see text] is an isomorphism for all [Formula: see text] and [Formula: see text] We also prove a vanishing result for relative negative [Formula: see text]-groups of a subintegral map.

Pinwheels and Quarter Turns

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ‎ denote the second iterate of the outer billiards map defined on R 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ‎, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ‎. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.

An upper bound for the Tarski numbers of nonamenable groups of piecewise projective homeomorphisms

The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. 110(12) (2013) 4524–4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn. 10(1) (2016) 177–200]. These groups do not contain non-abelian free subgroups. In this paper, we prove that the Tarski number of all groups in both families is at most 25. In particular, we demonstrate the existence of a paradoxical decomposition with 25 pieces. Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise [Formula: see text] homeomorphisms of [Formula: see text] with rational breakpoints and an affine map that is a not an integer translation.