# 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
$$H_{delta}^{d}(S)=inf left{sum_{i=1}^{infty}left(operatorname{diam} U_{i}right)^{d}: bigcup_{i=1}^{infty} U_{i} supseteq S, operatorname{diam} U_{i}, $$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$$.