Let R > 0 and a sequence of N variables: $m_1, m_2, \ldots, m_N$ taking values in [-2R,2R] be given. We would like to study the motion of a particle inside the square $\Omega$ = [0,R]², in particular, to study alternating horizontal-vertical motion emanating from a point $P_0$ belonging to $\Omega$. So we perform sequential translations in alternating horizontal and vertical sense according to the elements of our given sequence. At each of the N stages of motion, there are three possibilities:

1.      the translation ends in a point inside $\Omega$ (“no bounce”);

2.      the translation ends in a point on the boundary of $\Omega$ (“kiss”);

3.      the translation ends in a point exterior to $\Omega$ (“bounce”).

Naturally, in the “bounce” case, the motion of the particle is deflected off the boundary back into the interior of $\Omega$. At the end of each stage of motion, the particle then turns to the right with respect to the last direction followed. It is easy to see that the collection of initial points $P_k$ that result in a trajectory for which at least one stage of motion achieves a “kiss” forms a rectilinear partition of $\Omega$, which we shall refer to as $\Pi$. The rectangles constituting $\Pi$ we shall refer to as the cells of $\Pi$. Inherent to this construction are three kinds of phase transition:

1.      spatial phase transition: in this case, the displacement map $\Gamma$ which maps the initial phase space point $P_0$ to the end of the trajectory followed therefrom, varies discontinuously on segments connecting the interiors of adjacent cells;

2.      temporal phase transition: here, as the sequence $m_1, m_2, \ldots, m_N$ changes continuously beyond critical perturbation levels $\epsilon_j$, small supercritical perturbation results in a discontinuous change of $\Pi$;

3.      asymptotic phase transition: here, we consider the case that $N \rightarrow \infty$ and $R \rightarrow \infty$. Then we consider $D_h$ and $D_v$ to be the horizontal and vertical line densities in $\Pi(N,R)$. Supercritical perturbations of relative growth thresholds result in a positive value for $D_h$ and $D_v$, while subcritical perturbations result in a zero value for $D_h$ and $D_v$.

We will examine some special cases for the values of the motion variables, in particular, the taking of finite number of values and the resulting distribution of “kiss” points in $\Omega$. It is clear that all “kiss” lines are situated at distances of the form: linear combination of the motion variables. The coefficients of these linear combinations taken collectively over $\Omega$ forms a “coefficient space” whose size varies directly with the rational dependence of the motion variables and N. Similarly, is easy to see there is a relation between $D_h, D_v$ and the rational dependence of the motion variables. Finally, we will examine expected value of the displacement map $\Gamma$ and a formula for the fiber of a boundary point $P_b$ of $\Omega$.

Numerical experiments on the interaction between the large- and small-scale motion of the Navier-Stokes Equations

We show that the moduli space of real cubic surfaces has, in a natural way, the structure of real hyperbolic orbifold of dimension four. We discuss the structure of this space, its fundamental group, and we compute its exact hyperbolic volume. As a result we can, for instance, show that real cubics with twenty-seven real lines comprise less than two percent of the full space.