On first return path systems (Q5920595)

A trajetory is a sequence \((P_n)^\infty_{n=0}\) with the following properties: (i) \(P_0= 0\), \(P_1= 1\); (ii) \(P_n\neq P_m\) for \(n\neq m\); (iii) \(0\leq P_n\leq 1\) for all \(n\); (iv) \(\{P_n: n=0,1,\dots\}\) is dense in \([0,1]\). Let \(x\in [0,1]\). A path at \(x\) is a set \(R_x\subset [0,1]\) such that \(x\in R_x\) and \(x\) is a point of accumulation of \(R_x\). A system of path \(R\) is a collection \(\{R_x: x\in [0,1]\}\), where each \(R_x\) is a path at \(x\). Let \((P_n)\) be a fixed trajectory. For a given interval \((a,b)\subset [0,1]\), \(r(a,b)\) will be the first element of the trajectory in \((a,b)\). For \(0\leq y< 1\), the right first return path to \(y\), \(R^+_y\), is defined recursively via \(y^+_1= y\), \(y^+_2= 1\) and \(y^+_{k+1}= r(y, y^+_k)\) for \(k\geq 2\). For \(0< y\leq 1\), the left first return path to \(y\), \(R^-_y\), is defined similarly. For \(0< y< 1\), \(R_y= R^-_y\cup R^+_y\), and \(R_0= R^+_0\), \(R_1= R^-_1\). The path system \(R^+= \{R^+_x: x\in [0,1)\}\), \(R^-= \{R^-_x: x\in (0,1]\}\) and \(R= \{R_x: x\in (0,1)\}\cup\{R^+_0, R^-_1\}\) are called the right first return system, the left first return system and the first return system of path generated by \((P_n)\), respectively. The author deals with some properties introduced in quoted papers (the continuous system of path, the path system of congruent type, the internal intersection condition (IIC)). It is known that for a first return system of paths the right path system \(R^+\) (the left path system \(R^-\)) is right (is left) continuous and \(R\) satisfies IIC property. It is shown that any path system of congruent type \(R= \{R_x\}= \{x+ T: x\in [0,1]\}\) with \(T\) countable cannot have IIC property, thus it cannot contain any first return system of paths. The effect of turbulence on trajectories by treating them as sequences is also investigated.