Necessary properties of boundary degree sets of solutions to linear Pfaff systems (Q5951324)

Let us consider a completely integrable linear Pfaffian system \[ \begin{gathered} dx=A_1(t)dt_1 +A_2(t)dt_2\quad \bigl( x\in\mathbb{R}^n,\;t=(t_1,t_2) \in \mathbb{R}^2_{\geq 1},\\ \partial A_1/ \partial t_2+A_1A_2= \partial A_2/ \partial t_1+ A_2 A_1\bigr)\end{gathered} \] with bounded and continuously differentiable matrices \(A_1,A_2\). A lower characteristic vector \(p=p[x]\in \mathbb{R}^2\) and lower characteristic degree \(d=d_x(p)\in\mathbb{R}^2\) of a nontrivial solution \(x: \mathbb{R}^2_{\geq 1}\) \(\mathbb{R}^n/ \{0\}\) are defined by the conditions \[ \ln_x(p,d)=0,\;\ln_x(p,d+ \varepsilon e_i) <0\;\bigl(e_i=(2-i,i-1)\in\mathbb{R}^2;\;i=1,2;\text{ for all } \varepsilon >0\bigr) \] where \[ \ln_x(p,d)= \lim\inf_{t \to\infty} (\ln \bigl \|x(t)\bigr \|-(p,t)- \bigl(d,\ln t)\bigr)/\bigl \|\ln t\bigr\| \] \((\ln t=(\ln t_1,\ln t_2))\). Let \(P_x=\bigcup p[x]\) be the lower characteristic set. Then \(P_x=\{(p_1,\varphi (p_1)\}\), \(\alpha_x\leq p_1\leq \beta_x\), where \(\varphi: [\alpha_x, \beta_x]\to [a_x,b_x]\) is a certain monotone concave function. The authors introduce the lower degree set \(D_x(p) \subset\mathbb{R}^2\) involving all points \(d_x(p)\) and deal with the particular subcase of the (left) boundary lower degree set \(D_x(p)\) if \(p=(\alpha_x, b_x)=(\alpha_x,\varphi (\alpha_x))\) is the left boundary point of \(P_x\). In general, it can be empty. But there exist such Pfaffian equations that the left boundary lower degree set of any nontrivial solution is a nondecreasing concave curve \(\Gamma=\{d= (d_1,d_2)\}\) with the ranges \(-\infty< d_1<\infty\) and \(-\infty< d_2<0\). Then the geometry of such left boundary lower degree sets \(D_x(p)\) is thoroughly investigated.