Unit 5, Part I: Three-state conditional laws at arbitrary finite order

This part studies A §§2–3, pp.3–10. Fix a positive integer $n\ge2$ and set $I=\{1,\ldots,n\}$. The main claim concerns all $n$ positions, but the pointwise input on general auxiliary systems still uses at most three. We develop that shorter-average result and show how it determines three-state conditional marginals of the selected law.

A fixed finite tuple and all its integer translates involve only countably many coordinates, so the standard Borel factor reduction remains available. The standard-factor supplement addresses completion and the pullback of Borel convergence sets.

There are two external inputs: smooth triple pointwise convergence on general systems, A Proposition 2.2, and the relative structure and nonergodic distal pointwise results in Proposition 2.3. These connect respectively to C's smooth oscillation estimate and the structural theorems stated in Unit 4. The deduction below is conditional on those precise statements. It does not use the all-orders mixing claim in A Theorem 2.1.

1. First establish a common $L^2$ limit on a general system

Source: A Lemma 2.4 and the following paragraph, p.5.

Let $(Y,m,R)$ be a standard invertible probability-preserving system and $W$ its maximal measure-distal factor. A Proposition 2.3 asserts that $Y\to W$ is relatively weakly mixing. For pairwise distinct positive integers $a_1,\ldots,a_d$ and bounded functions $G_1,\ldots,G_d$, set

The relative norm-replacement formula proved in detail in Unit 4 is precisely A Lemma 2.4:

The proof allows distinct nonzero integer slopes, including negative ones. Positive slopes are used here to apply the distal pointwise theorem in A Proposition 2.3(5).

Set $a=\max_j a_j$. At position $a_j$ place $\mathbb E[G_j\mid W]$ and the constant $1$ elsewhere. This writes $B_N$ as an average on the distal system with slopes $1,\ldots,a$. Hence $B_N$ converges almost everywhere. All $B_N$ have a common finite bound, so dominated convergence gives $B_N\to L$ in $L^2$. By (A-L2),

Here $d$ may be any fixed finite number, but the conclusion is norm convergence only. On a general system, $L$ need not be constant or equal to a product of integrals. On the original mixing system, the trivial factor already satisfies relative weak mixing, giving

2. The precise smooth triple input

Source: A Proposition 2.2, pp.3–4, based on C Proposition 3.5 and Lemmas 3.6 and 3.9.

Take three pairwise distinct nonzero integers $b_1,b_2,b_3$ satisfying $\gcd(|b_1|,|b_2|,|b_3|)=1$. For every standard invertible probability-preserving system $(Y,m,S)$ and three bounded functions $g_j$, the proposition asserts almost-everywhere convergence of

Here $\varphi\in C_c^\infty(\mathbb R;\mathbb R)$ satisfies

and for each fixed $\varphi$ there is a constant $G\ge1$ such that

For every $c\in[1,2]$ and every $\varepsilon>0$, there also exists such a weight $\varphi$ with

These weights must change sign, and their supports may depend on $\varepsilon$ during approximation. Unit 1 develops the moment correction and the changing support explicitly. At present, only convergence of each smooth average is used; no formula for its limit on a general system is asserted.

3. A Lemma 2.5: At most three factors converge through all integer lengths

The statement is on p.5, with the proof continuing on p.6.

Let $(Y,m,R)$ be a standard invertible probability-preserving system, let $1\le d\le3$, and let $a_1,\ldots,a_d$ be pairwise distinct positive integers. For fixed bounded functions $G_j$, the conclusion is

Neither ergodicity nor mixing is assumed.

3.1 Completing the triple and normalizing

If $d<3$, choose additional positive integer slopes distinct from those already present and use the function $1$ in the new positions. The product is unchanged and now has three distinct positive slopes. Let their greatest common divisor be $g$, write $gb_1,gb_2,gb_3$, and set $S=R^g$. Divide each nonzero function by its supremum norm. It suffices to treat

If a function is zero, convergence is immediate.

3.2 One countable choice of smooth averages

For every rational $c\in[1,2]$ and integer $r\ge1$, choose an admissible $\varphi_{c,r}$ in advance such that

There are countably many choices. Intersect their full-measure smooth-convergence sets to obtain a fixed full-measure set $E_0$. All later changes of $c,r$ are handled on the same $E_0$.

Set $M_k(c)=\lfloor c2^k\rfloor$. The discrete sum for the interval weight is exactly

It differs from $A_{M_k(c)}$ only in normalization. Hence

The second term is a Riemann sum for a fixed compactly supported function with finitely many jumps, and tends to $\|h_c-\varphi_{c,r}\|_1$.

3.3 This proves the Cauchy property without a prescribed smooth limit

Fix $y\in E_0$, rational $c$, and $r$. Smooth convergence and (Smooth-compare) imply

Let $r\to\infty$. The tail diameter is zero, so $A_{M_k(c)}(y)$ converges.

The argument only uses convergence for each fixed smooth weight. It does not assume that different $\varphi$ have the same limit. On a general nonmixing system, the product-limit identification from Unit 1 cannot simply be imported.

3.4 The common norm limit identifies every rational-ratio limit

Section 1 gives $A_N\to L$ in $L^2(m)$. Fix rational $c$ and denote the almost-everywhere limit just obtained by $L_c$. By $|A_N|\le1$ and dominated convergence, $A_{M_k(c)}\to L_c$ in $L^2$. The same subsequence converges to $L$, so $L_c=L$ almost everywhere.

Intersect once more over the countably many rational $c$ to obtain $E\subseteq E_0$. On $E$, all these subsequences converge to the same chosen representative of $L$.

3.5 A finite rational grid fills the gaps between lengths

For any sequence satisfying $|F_l|\le1$, if $M\ge N$, then

Interchanging $M,N$ gives A (2.3).

Fix $\eta>0$. Choose a finite rational grid containing the endpoints $1,2$, denoted $F_\eta$, such that every point of $[1,2]$ lies within $\eta$ of a grid point. If $2^k\le N<2^{k+1}$, choose $c\in F_\eta$ satisfying $|N/2^k-c|\le\eta$. Then

On the fixed set $y\in E$, the finitely many grid subsequences converge simultaneously to $L(y)$. Thus

Let $\eta\downarrow0$ to obtain convergence through all lengths. Every grid uses rational numbers already covered, so no uncountable intersection of full-measure sets occurs.

4. Arbitrary-order failure leaves a nonempty centered support

Source: A §3.1, p.6.

Suppose an $n$-fold average on the original mixing system fails to converge pointwise to the product of the integrals. Split each function into its mean and mean-zero part, obtaining finitely many subset terms. The constant term is the proposed limit. If every nonempty centered subset average tended to zero almost everywhere, a finite full-measure intersection would give convergence of the original average. Some nonempty support $J_0\subseteq I$ must therefore still witness failure.

Splitting real and imaginary parts and rescaling gives real mean-zero functions $f_i$, $\|f_i\|_\infty\le1$, for which

does not tend to zero on a positive-measure set. Ordinary mixing still gives $B_L\to0$ in $L^2$. Choose positive or negative deviations of the bounded real sequence, changing one function's sign if necessary. There are $\delta>0$ and $\mu(E_*)>0$ such that, for every $x\in E_*$, arbitrarily large $L$ satisfy $B_L(x)>2\delta$.

Section 3 and the mixing norm limit exclude $|J_0|\le3$, so $|J_0|\ge4$. Unlike the fourfold case, we cannot yet assert $J_0=I$: no pointwise induction hypothesis for every shorter average has been used. This unknown support size is the starting point for the minimal-support argument in the next part.

5. Phase selection does not require the first two positions to belong to the support

Source: A Lemma 3.1, p.7, equations (3.3)–(3.6).

Using (2.3), choose a sufficiently fine finite ratio grid $F\subseteq[1,2]$ containing $1$. For each $a$ and $x\in E_*$, select the first pair among the countably many $(c,k)\in F\times\{a,a+1,\ldots\}$ satisfying $B_{\lfloor c2^k\rfloor}(x)>\delta$, and set $N_a(x,u)=c2^k$. Outside $E_*$, use $N_a=2^a$. The selection is measurable and independent of the phase $u$.

In the unit-roof suspension flow

write the spatial coordinate as $\xi$ and the phase as $\upsilon$. Choose nonzero continuous nonnegative circle functions $\alpha,\beta$ supported in $(1/4,1/2)$ and $(0,1/(4n))$, respectively. Define

When $t=l+s$ and $0\le s<1$, the two circle differences are $u,s$. A nonzero phase weight therefore implies

Hence $\xi(S_C^{it}(x,u))=T^{il}x$. The weight is defined on the full phase array, although its formula uses only phase coordinates $1,2$. Those coordinates still exist even if $1,2\notin J_0$, so the weight remains defined.

For any real $L\ge2$, complete unit-time intervals contribute the fixed factor $\int\beta$, while the incomplete interval, endpoint changes, and normalization differences cost $O(L^{-1})$. More precisely, set $M=\lfloor L\rfloor$. First obtain

Changing the sum over $0,\ldots,M-1$ to the sum over $1,\ldots,M$ costs at most $2$. Replacing $M/L$ by $1$ costs at most $1/L$. The original time average is therefore

with error uniform in $x,u,L$.

On $E_*$, take $L=N_a$. The main term has integral at least

On the complement, $N_a=2^a$ and $B_{2^a}\to0$ in $L^2$, so its integral tends to zero. The bound $N_a\ge2^a$ makes the error $O(2^{-a})$. This proves the strictly positive limit inferior in the paper's equation (3.4).

6. The selected law of $n$ states

Source: A Proposition 3.2, p.8.

Fix all the selectors $N_a$ above and a free ultrafilter $\mathcal U$. For each standard flow extension $\pi_Y:Y\to C$, the probability construction of Unit 4 applies without change to any fixed finite number of positions. It yields a unique probability measure $\lambda_Y$ on $Y^I$ satisfying

for all fixed bounded measurable $G_i$.

No new compactness requirement arises beyond the four-coordinate argument. The product error is bounded by $\sum_{i=1}^n$ one-coordinate $L^1$ errors, with $n$ fixed. The same calculations give one-coordinate marginals $m_Y$, invariance under $(y_i)\mapsto(S_Y^{it}y_i)$, for every extension $\rho:Y'\to Y$ the compatibility identity $(\rho^I)_*\lambda_{Y'}=\lambda_Y$, and density of stage functions in $L^2$ of the limiting selected law along countable inverse limits. See Part I of Unit 4 for the probability construction, continuous approximation, and inverse-limit realization.

Flow-factor relations hold modulo null sets at each fixed real time. The countable inverse limit uses the full product space and coordinatewise action. Its measure is concentrated on compatible sequences, but that compatibility set need not be pointwise invariant simultaneously at all real times.

7. Selection does not change the full parameter law

Source: A Proposition 3.3, p.9, equations (3.8)–(3.9).

Let $\omega_Y:Y\to W_Y$ be the maximal distal factor of the time-one map, with conditional measures $m_{Y,w}$. Write

Every time map of the original flow preserves this factor. All time integrals are evaluated on the original jointly Borel flow $Y$.

For parameter functions $H_i$, write $t=l+s$. At each fixed $s$, the functions

are $W_Y$-measurable function classes. Their time-one averages converge by the distal pointwise theorem at every finite order. Fubini gives convergence for almost every $y$ at almost every $s$. Dominated convergence over $s\in[0,1)$ then gives convergence of the continuous average through all real lengths. The endpoint unit-interval errors vanish.

The selected lengths are at least $2^a$, so the same pointwise limit holds along them. Bounded convergence and (Selected-n) give

This is the deterministic-length joint parameter law. It does not assert that the parameters are mutually independent.

8. At most three states are independent conditional on all parameters

Source: A Proposition 3.3, pp.9–10, equations (3.10)–(3.12). The full conclusion is that every nonempty $J\subseteq I$ with $|J|\le3$ satisfies

8.1 First condition only on these positions' own parameters

The pointwise theorem in §3 and the continuous-time argument above identify the selected joint law of at most three states with its deterministic-length limit. At deterministic lengths, use relative norm replacement (A-L2) and integrate over the time remainder $s$. This gives

Conditional expectation commutes with a fixed time shift because that shift preserves $W_Y$. For each fixed $s$, the relative norm error tends to zero and has a common bound, so its integral over $s$ also tends to zero.

Multiply each $G_i$ by an arbitrary bounded parameter function $H_i(P_i)$. Product tests determine the measure, giving

8.2 Extend the conditioning to all parameters

The tower property alone cannot upgrade (Own-parameters) to (Three-slots). Relative disjointness is needed.

Use the law of $B_J=(P_i)_{i\in J}$ as the common base. Under the product transformation with speeds $i$, the state system in (Own-parameters) is a conditional product of relatively weakly mixing extensions, hence relatively weakly mixing over $B_J$. The full parameter system $P$ is a joining of the distal systems $W_Y$ at integer speeds. It is therefore distal and relatively distal over its factor $B_J$.

Their joint law under $\lambda_Y$ is an invariant joining identifying the common base. A Proposition 2.3(4) forces relative independence over $B_J$. Thus adding the remaining parameters does not change the state conditional law, proving (Three-slots).

These laws can be realized as simultaneous kernel identities. For fixed finite $n$, the condition $|J|\le3$ permits only finitely many subsets $J$. Choose countable Borel generating algebras for the state and parameter spaces. First take a common full-measure parameter set for all generating tests, then extend to every Borel set by uniqueness of measures. The result is equality of conditional probability measures on one parameter full-measure set, not merely separate identities modulo null sets for individual tests.

When $n>4$, a proper subset may contain four or more positions. The argument gives no product law for those complete states. The next part recovers higher-order conditional products only for specified spatial lists through minimal support. That must not be rewritten as independence of arbitrary complete states.

9. The normalized flow and retained positive correlation

Source: A Lemmas 3.4–3.5, p.10.

For $b\in I$, set

Time rescaling preserves invariance of the selected law. Position $b$ has speed one and marginal $m_Y$, so its projection is a flow-extension map. The old parameter array $P$ is a joining of distal factors at rational speeds $i/b$ and therefore lies in the maximal distal factor of the new time-one map.

For $Y=C$, uniformly approximate the phase weight $\Psi$ by finite sums of products of continuous one-coordinate functions. Equation (Selected-n) and the strictly positive limit inferior in §5 give

The suspension time-one map preserves phase, so $\Psi$ is a function of the full distal parameter array. This supplies the initial nonzero correlation for the minimal-support choice in the next part.

Scope in the original text

• A pp.3–5, Propositions 2.2–2.3 and Lemma 2.4: the smooth input, relative structure, and common norm limit.
• A pp.5–6, Lemma 2.5: full-length pointwise convergence for at most three factors on general systems.
• A pp.6–8, Lemma 3.1 and Proposition 3.2: nonempty centered support, positive phase-weighted correlation, and the selected law of $n$ states.
• A pp.9–10, Proposition 3.3 and Lemmas 3.4–3.5: the full parameter law, conditional laws for at most three states, and the normalized flow.

Auxiliary-group preservation, minimal support, and the maximal auxiliary-position count in A §§4–6 have not yet been discussed. In particular, the three-state conditional law has not been extended to every proper subset of states.