real analysis – Uniformly estimate for hausdorff dimension of section of a given flow

I wondering weather the following result
is right:

$I=(0.1)$ . consider a continuous dynamic system,

begin{aligned}
&varphi_{t}: I^{2} rightarrow I^{2} \
&(x, y) rightarrow({t x}, y)
end{aligned}

where ${x}$ is the part of $tx$. we have $forall t,sin R, varphi_{t+s}=varphi_{t} circ varphi_{s}$ , and consider following two maps:
begin{array}{r}
P: I^{2} longrightarrow I Q: I^{2} rightarrow I\
quad(x, y) rightarrow{x+y} (x, y) rightarrow x
end{array}

Let$quad S subseteq I^2$ be a subset of $I^2$, satisfied $mu_{d+1}^{*}left(cup_{t in(0 . 1)}varphi_{t}(S))=0right.$ ($1leq dleq 1$, $mu_{d+1}^{*}$ is the $d+1$– hausforff outer measure), and $mu^{*}(Q(S))>0$ ($mu^{*}$ is the lebesgue outer measure)
then for almost all $t in(0.1)$ except a lebesgue measure null set, we have
$mu_{d}^{*}left(Pleft(varphi_{t}(s)right))=0right.$

Let $S$ be any subset of $I^2,$ and $delta>0$ a real number. Define
$H_{delta}^{d}(S)=inf left{sum_{i=1}^{infty}left(operatorname{diam} U_{i}right)^{d}: bigcup_{i=1}^{infty} U_{i} supseteq S, operatorname{diam} U_{i}<deltaright}$, $H^{d}(S):=sup _{delta>0} H_{delta}^{d}(S)=lim _{delta rightarrow 0} H_{delta}^{d}(S)$ is the $d$-hausdorff outer measure of $S$.