SilverSight/docs/fundamental_math/G1_CHAOS_GAME_CONTRACTION.md

Theorem G1: Contraction of the Chaos Game on the Fisher–Rao Simplex

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Domain note: All Fisher metric computations require p_i > 0 for all i. Raw byte frequencies produce p_i = 0 for unseen classes. Apply Laplace smoothing: F(E)_i = (f_i + 1)/(|E| + 8) to ensure strict positivity. This keeps all computations in the open simplex \Delta_7 and regularizes the Fisher metric near boundaries.

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1. Setup

1.1 The open probability simplex

Let N = 8. The open probability simplex is

\Delta_{N-1} = \Delta_7 = \Bigl\{p = (p_1,\dots,p_N) \in \mathbb{R}^N : p_i > 0,\; \sum_{i=1}^N p_i = 1\Bigr\}.

Its tangent space at p \in \Delta_7 is

T_p\Delta_7 = \Bigl\{v = (v_1,\dots,v_N) \in \mathbb{R}^N : \sum_{i=1}^N v_i = 0\Bigr\}.

The simplex is an $(N-1)$-dimensional smooth manifold diffeomorphic to \mathbb{R}^{N-1}.

1.2 The Fisher metric and Fisher distance

The Fisher information metric (Amari 1985, §2; also Rao 1945) is the Riemannian metric on \Delta_7 defined by

g_p(u,v) = \sum_{i=1}^N \frac{u_i v_i}{p_i}, \qquad u,v \in T_p\Delta_7.

The Fisher distance d_F is the geodesic distance induced by g:

d_F(p,q) = \inf_{\gamma} \int_0^1 \sqrt{g_{\gamma(t)}\bigl(\gamma'(t),\gamma'(t)\bigr)}\,dt,

where the infimum is over all piecewise smooth curves \gamma: [0,1] \to \Delta_7 with \gamma(0)=p, \gamma(1)=q.

1.3 The \sqrt{p} embedding into the sphere \mathcal{S}^{N-1}

Let \mathcal{S}^{N-1} = \{x \in \mathbb{R}^N : \|x\|_2 = 1\} denote the unit Euclidean sphere, and let \mathcal{S}^{N-1}_+ = \mathcal{S}^{N-1} \cap (0,\infty)^N denote its positive orthant. Define the smooth embedding

\Phi : \Delta_7 \longrightarrow \mathcal{S}^{N-1}_+, \qquad \Phi(p) = \sqrt{p} = (\sqrt{p_1},\dots,\sqrt{p_N}).

The differential of \Phi at p in direction v \in T_p\Delta_7 is

d\Phi_p(v) = \Bigl(\frac{v_1}{2\sqrt{p_1}},\dots,\frac{v_N}{2\sqrt{p_N}}\Bigr).

1.4 Pullback of the round metric — the S^7 embedding theorem

Equip \mathcal{S}^{N-1} with the round metric g_{\mathcal{S}} induced from the Euclidean inner product on \mathbb{R}^N. The pullback metric \Phi^* g_{\mathcal{S}} satisfies

(\Phi^* g_{\mathcal{S}})_p(u,v) = \langle d\Phi_p(u),\, d\Phi_p(v) \rangle_{\mathbb{R}^N} = \sum_{i=1}^N \frac{u_i v_i}{4p_i} = \frac{1}{4}\, g_p(u,v).

Hence

\boxed{g_p(u,v) = 4\, (\Phi^* g_{\mathcal{S}})_p(u,v)}

and at the level of distances,

\boxed{d_F(p,q) = 2\, d_{\mathrm{arc}}\bigl(\Phi(p),\, \Phi(q)\bigr)}

where d_{\mathrm{arc}}(x,y) = \arccos(\langle x,y \rangle) is the geodesic (arc) distance on \mathcal{S}^{N-1}.

Proof of the distance formula. Let \gamma be a geodesic from p to q for g. Then \tilde\gamma = \Phi \circ \gamma is a curve on \mathcal{S}^{N-1}_+ from \Phi(p) to \Phi(q), and

\int_0^1 \sqrt{g_{\gamma}(\gamma',\gamma')}\,dt = \int_0^1 2\sqrt{g_{\mathcal{S}}(\tilde\gamma',\tilde\gamma')}\,dt = 2\, \mathrm{Length}(\tilde\gamma) \geq 2\, d_{\mathrm{arc}}(\Phi(p),\Phi(q)).

Conversely, the geodesic on \mathcal{S}^{N-1} between \Phi(p) and \Phi(q) lies in \mathcal{S}^{N-1}_+ (since the positive orthant is convex on the sphere), and its inverse image under \Phi is a geodesic for g. The factor of 2 follows from the metric identity above. \square

1.5 Bhattacharyya and chordal distances on \mathcal{S}^{N-1}

The Bhattacharyya distance on \Delta_7 is defined by

d_B(p,q) = \arccos\bigl(BC(p,q)\bigr), \qquad BC(p,q) = \sum_{i=1}^N \sqrt{p_i q_i}.

Since \langle \Phi(p), \Phi(q) \rangle = BC(p,q), we have immediately

\boxed{d_B(p,q) = d_{\mathrm{arc}}\bigl(\Phi(p),\, \Phi(q)\bigr)}.

The chordal distance on \mathcal{S}^{N-1} is the Euclidean distance in \mathbb{R}^N:

d_{\mathrm{chord}}(x,y) = \|x-y\|_2 = \sqrt{2 - 2\langle x,y \rangle} = 2\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr).

The following elementary comparison holds for all x,y \in \mathcal{S}^{N-1}_+ (since d_{\mathrm{arc}} \in [0,\pi/2] in the positive orthant):

\frac{2}{\pi}\, d_{\mathrm{arc}}(x,y) \leq d_{\mathrm{chord}}(x,y) \leq d_{\mathrm{arc}}(x,y).

In particular, on \Delta_7:

\frac{1}{\pi}\, d_F(p,q) \leq d_{\mathrm{chord}}(\Phi(p),\Phi(q)) \leq \frac{1}{2}\, d_F(p,q).

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2. The Maps and the Chentsov Condition

2.1 Affine stochastic maps on \Delta_7

Let k \geq 2. For each j \in \{1,\dots,k\}, let A_j be an N \times N matrix and b_j \in \mathbb{R}^N a vector satisfying:

(H1) A_j is stochastic: (A_j)_{i\ell} \geq 0 for all i,\ell, and \sum_{i=1}^N (A_j)_{i\ell} = 1 for all \ell (column-stochastic; equivalently, A_j^\top \mathbf{1} = \mathbf{1}).

(H2) (b_j)_i \geq 0 for all i, and B_j := \sum_{i=1}^N (b_j)_i > 0.

The Chentsov monotonicity condition requires that each A_j is induced by a sufficient statistic (coarse-graining). Concretely:

(C) For each j, there exists a surjective map T_j: \{1,\dots,N\} \to \{1,\dots,m_j\} with m_j < N such that

(A_j)_{i\ell} = \mathbf{1}_{\{T_j(\ell) = i\}}.

In other words, A_j is a deterministic channel: each input \ell is mapped deterministically to the output T_j(\ell). Equivalently, each column of A_j contains exactly one entry equal to 1 and the rest are 0.

Under (H1) and (H2), for p \in \Delta_7 we have \sum_i (A_j p + b_j)_i = 1 + B_j, so the normalized affine map

w_j(p) = \frac{A_j p + b_j}{1 + B_j}

satisfies \sum_i w_j(p)_i = 1 and w_j(p)_i > 0; thus w_j: \Delta_7 \to \Delta_7 is well-defined. Since A_j has nonnegative entries and b_j \geq 0, the map w_j sends \Delta_7 into the compact subset

\Delta_7^{(\varepsilon_j)} := \Bigl\{p \in \Delta_7 : p_i \geq \frac{(b_j)_i}{1+B_j}\Bigr\}, \qquad \varepsilon_j = \min_i \frac{(b_j)_i}{1+B_j} > 0.

Each w_j extends continuously to the closed simplex \overline{\Delta}_7.

2.2 Induced map on \mathcal{S}^{N-1}_+

Via the \sqrt{p} embedding, w_j induces the map \widehat{w}_j : \mathcal{S}^{N-1}_+ \to \mathcal{S}^{N-1}_+ defined by

\widehat{w}_j(x) = \Phi\bigl(w_j(x^2)\bigr) = \frac{\sqrt{A_j x^2 + b_j}}{\sqrt{1+B_j}},

where x^2 = (x_1^2,\dots,x_N^2) and the square root in the numerator is componentwise.

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3. Lemma 1 — Contraction on the Sphere

Lemma 1. Let w(p) = (Ap + b)/(1+B) satisfy (H1), (H2), and (C). Then the induced map \widehat{w}: \mathcal{S}^{N-1}_+ \to \mathcal{S}^{N-1}_+ satisfies

d_{\mathrm{chord}}\bigl(\widehat{w}(x),\, \widehat{w}(y)\bigr) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{chord}}(x,y) \qquad \forall\, x,y \in \mathcal{S}^{N-1}_+,

and consequently

d_B\bigl(w(p),\, w(q)\bigr) \leq \frac{1}{\sqrt{1+B}}\, d_B(p,q) \qquad \forall\, p,q \in \Delta_7.

Proof. Define F: \mathbb{R}^N_+ \to \mathbb{R}^N by

F_i(z) = \frac{\sqrt{(Az^2)_i + b_i}}{\sqrt{1+B}}, \qquad i = 1,\dots,N,

so that \widehat{w}(x) = F(x) for x \in \mathcal{S}^{N-1}_+. The differential of F at z in direction \xi \in \mathbb{R}^N is

dF_z(\xi)_i = \frac{1}{\sqrt{1+B}} \cdot \frac{(A(z\xi))_i}{\sqrt{(Az^2)_i + b_i}},

where (z\xi)_\ell = z_\ell \xi_\ell. Therefore,

\|dF_z(\xi)\|_2^2 = \frac{1}{1+B} \sum_{i=1}^N \frac{\bigl(\sum_{\ell=1}^N A_{i\ell} z_\ell \xi_\ell\bigr)^2}{\sum_{\ell=1}^N A_{i\ell} z_\ell^2 + b_i}.

For each i, apply the Cauchy–Schwarz inequality to the numerator:

\Bigl(\sum_{\ell} A_{i\ell} z_\ell \xi_\ell\Bigr)^2 = \Bigl(\sum_{\ell} \sqrt{A_{i\ell}} \cdot \sqrt{A_{i\ell}}\, z_\ell \xi_\ell\Bigr)^2 \leq \Bigl(\sum_{\ell} A_{i\ell}\Bigr)\Bigl(\sum_{\ell} A_{i\ell} z_\ell^2 \xi_\ell^2\Bigr).

Since A is column-stochastic, \sum_i A_{i\ell} = 1 for each \ell, but here we need \sum_\ell A_{i\ell}. Because A is a deterministic channel matrix (condition (C)), each row of A corresponds to a fiber of T, and \sum_\ell A_{i\ell} = |T^{-1}(i)| counts the size of the $i$-th fiber. In particular, \sum_\ell A_{i\ell} \geq 1.

However, we use the sharper bound: since (Az^2)_i + b_i \geq (Az^2)_i,

\frac{\bigl(\sum_{\ell} A_{i\ell} z_\ell \xi_\ell\bigr)^2}{\sum_{\ell} A_{i\ell} z_\ell^2 + b_i} \leq \frac{\bigl(\sum_{\ell} A_{i\ell} z_\ell^2\bigr)\bigl(\sum_{\ell} A_{i\ell} \xi_\ell^2\bigr)}{\sum_{\ell} A_{i\ell} z_\ell^2 + b_i} \leq \sum_{\ell} A_{i\ell} \xi_\ell^2,

where the second inequality uses \frac{s \cdot t}{s + b_i} \leq t for s,t \geq 0 and b_i \geq 0, with s = \sum_\ell A_{i\ell} z_\ell^2 and t = \sum_\ell A_{i\ell} \xi_\ell^2.

Summing over i:

\|dF_z(\xi)\|_2^2 \leq \frac{1}{1+B} \sum_{i=1}^N \sum_{\ell=1}^N A_{i\ell} \xi_\ell^2 = \frac{1}{1+B} \sum_{\ell=1}^N \xi_\ell^2 \sum_{i=1}^N A_{i\ell} = \frac{\|\xi\|_2^2}{1+B},

where the last equality uses \sum_i A_{i\ell} = 1 (column-stochastic).

Now restrict to the tangent space of \mathcal{S}^{N-1}. For x,y \in \mathcal{S}^{N-1}_+, the geodesic segment [x,y] lies in the positive orthant. By the mean value theorem on \mathcal{S}^{N-1} (using the exponential map or working in ambient \mathbb{R}^N with the constraint \|z\|=1):

\|\widehat{w}(x) - \widehat{w}(y)\|_2 = \|F(x) - F(y)\|_2 \leq \sup_{z \in [x,y]} \|dF_z|_{T_z\mathcal{S}^{N-1}}\|_{\mathrm{op}} \cdot \|x-y\|_2.

For \xi \in T_z\mathcal{S}^{N-1} (i.e., \langle z, \xi \rangle = 0), the bound \|dF_z(\xi)\|_2 \leq \|\xi\|_2/\sqrt{1+B} holds (the tangent constraint does not affect the estimate). Hence

d_{\mathrm{chord}}(\widehat{w}(x),\widehat{w}(y)) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{chord}}(x,y).

For the Bhattacharyya distance, recall that d_B = d_{\mathrm{arc}} and d_{\mathrm{chord}} = 2\sin(d_{\mathrm{arc}}/2). The chordal contraction yields

2\sin\Bigl(\frac{d_{\mathrm{arc}}(\widehat{w}(x),\widehat{w}(y))}{2}\Bigr) \leq \frac{1}{\sqrt{1+B}} \cdot 2\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr).

Applying \arcsin to both sides and using that \arcsin is convex on [0,1] with \arcsin(0)=0 (hence \arcsin(\lambda s) \leq \lambda \arcsin(s) for \lambda \in [0,1]):

d_{\mathrm{arc}}(\widehat{w}(x),\widehat{w}(y)) \leq 2\arcsin\Bigl(\frac{1}{\sqrt{1+B}}\sin\Bigl(\frac{d_{\mathrm{arc}}(x,y)}{2}\Bigr)\Bigr) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{arc}}(x,y).

Therefore

d_B(w(p),w(q)) = d_{\mathrm{arc}}(\widehat{w}(\Phi(p)),\widehat{w}(\Phi(q))) \leq \frac{1}{\sqrt{1+B}}\, d_{\mathrm{arc}}(\Phi(p),\Phi(q)) = \frac{1}{\sqrt{1+B}}\, d_B(p,q).

\square

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4. Lemma 2 — Equivalence of Distances

Lemma 2. For all p,q \in \Delta_7,

\boxed{d_F(p,q) = 2\, d_B(p,q)}.

Consequently, any map w: \Delta_7 \to \Delta_7 that is $\lambda$-Lipschitz in the Bhattacharyya distance d_B is also $\lambda$-Lipschitz in the Fisher distance d_F with the same contraction factor.

Proof. From §1.4, the \sqrt{p} embedding satisfies \Phi^* g_{\mathcal{S}} = \frac{1}{4} g, so the length of any curve is scaled by 1/2 when measured with g_{\mathcal{S}} versus g. The geodesic distance therefore satisfies d_F = 2\, d_{\mathrm{arc}} \circ (\Phi \times \Phi) = 2\, d_B.

If d_B(w(p),w(q)) \leq \lambda \, d_B(p,q) for all p,q, then

d_F(w(p),w(q)) = 2\, d_B(w(p),w(q)) \leq 2\lambda\, d_B(p,q) = \lambda\, d_F(p,q).

\square

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5. Proof of Theorem G1

Theorem G1. Let \{w_1,\dots,w_k\} be the IFS on \Delta_7 defined in §2.1, with each w_j satisfying (H1), (H2), and the Chentsov monotonicity condition (C). Then each w_j is a strict contraction on (\Delta_7, d_F) with

d_F\bigl(w_j(p),\, w_j(q)\bigr) \leq \lambda_j \, d_F(p,q), \qquad \lambda_j = \frac{1}{\sqrt{1+B_j}} < 1,

where B_j = \sum_i (b_j)_i > 0. Consequently, the IFS is contractive with global Lipschitz constant

\lambda = \max_{1 \leq j \leq k} \lambda_j = \max_j \frac{1}{\sqrt{1+B_j}} < 1.

Proof. Lemma 1 shows that each w_j is $\lambda_j$-Lipschitz in the Bhattacharyya distance d_B with \lambda_j = 1/\sqrt{1+B_j}. By hypothesis (H2), B_j > 0, so \lambda_j < 1. Lemma 2 shows that the same Lipschitz constant carries over to the Fisher distance d_F. Therefore each w_j is a strict contraction on (\Delta_7, d_F), and the IFS is contractive with constant \lambda = \max_j \lambda_j < 1. \square

5.1 Explicit bound in terms of singular values

The contraction factor can also be expressed in terms of the singular values of A_j. Define the nonlinear distortion of w_j at p as the operator norm

\kappa_j(p) = \sup_{v \in T_p\Delta_7 \setminus \{0\}} \frac{\|dw_j|_p(v)\|_{F,w_j(p)}}{\|v\|_{F,p}}.

A direct computation (as in the proof of Lemma 1, but pulled back to \Delta_7) gives:

\|dw_j|_p(v)\|_{F,w_j(p)}^2 = \frac{1}{1+B_j} \sum_{i=1}^N \frac{(A_j v)_i^2}{(A_j p + b_j)_i}.

Using the Fisher-score u_\ell = v_\ell/p_\ell and the variance inequality (Jensen gap) for each fiber of T_j:

\sum_{i} \frac{(A_j v)_i^2}{(A_j p)_i} = \|v\|_{F,p}^2 - \sum_{i} (A_j p)_i \cdot \mathrm{Var}_i(u),

where \mathrm{Var}_i(u) \geq 0 is the variance of the scores u_\ell over the fiber T_j^{-1}(i) with weights p_\ell/(A_j p)_i. The b_j term contributes the additional factor 1/(1+B_j). Hence:

\kappa_j(p) \leq \frac{1}{\sqrt{1+B_j}} \leq 1 - \frac{B_j}{2(1+B_j)}.

For the global Lipschitz constant, integrating along geodesics gives \mathrm{Lip}(w_j) = \sup_p \kappa_j(p) = 1/\sqrt{1+B_j}.

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6. Corollary — Chaos Game Convergence

Corollary (Hutchinson–Barnsley). The IFS \{w_1,\dots,w_k\} on (\Delta_7, d_F) satisfies the hypotheses of the Hutchinson–Barnsley theorem. Therefore:

(a) There exists a unique nonempty compact set \mathcal{A} \subset \overline{\Delta}_7, called the attractor, such that

\mathcal{A} = \bigcup_{j=1}^k w_j(\mathcal{A}).

(b) For any starting point p_0 \in \Delta_7, the chaos game sequence

p_{n+1} = w_{J_n}(p_n), \qquad n \geq 0,

where (J_n)_{n \geq 0} are i.i.d. uniform on \{1,\dots,k\}, converges to \mathcal{A} almost surely in the Hausdorff sense: d_F(p_n, \mathcal{A}) \to 0 a.s.

*(c) The Markov operator W defined by W(\mu) = \frac{1}{k}\sum_{j=1}^k (w_j)_\# \mu has a unique invariant probability measure \mu_* supported on \mathcal{A}, and W^n(\mu) \to \mu_* weakly for any initial probability measure \mu on \Delta_7.

Proof. By Theorem G1, \mathrm{Lip}(w_j) \leq \lambda < 1 for all j on the complete metric space (\Delta_7, d_F). The maps w_j extend continuously to the compactification \overline{\Delta}_7. The conclusions (a)–(c) are the standard Hutchinson–Barnsley theorem (Barnsley 1988, Theorem 3.1; Hutchinson 1981, §3.1) for contractive IFS on complete metric spaces. The existence and uniqueness of \mathcal{A} follows from the Banach fixed-point theorem applied to the set-map \mathcal{W}(K) = \cup_j w_j(K) on the space of nonempty compact subsets with the Hausdorff metric. Almost sure convergence of the chaos game follows from the contraction property and the law of large numbers for the coding space (Barnsley & Demko 1985). \square

6.1 The Open Set Condition

The IFS \{w_1,\dots,w_k\} satisfies the open set condition (OSC) on \Delta_7 if there exists a nonempty open set U \subset \Delta_7 such that

w_j(U) \subset U \quad \text{for all } j, \qquad \text{and} \qquad w_j(U) \cap w_\ell(U) = \varnothing \quad \text{for } j \neq \ell.

Sufficient condition for OSC. If the coarse-graining maps T_1,\dots,T_k have pairwise disjoint active domains — meaning the sets of coordinates that are not pointwise fixed satisfy

\{\ell : T_j(\ell) \neq \ell\} \cap \{\ell : T_\ell(\ell) \neq \ell\} = \varnothing \quad \text{for } j \neq \ell,

then the IFS satisfies the OSC on \Delta_7 with U = \Delta_7.

Proof sketch. Under the disjoint-active-domain condition, each w_j modifies a disjoint subset of coordinates. For the example in §7, each w_j only affects the pair S_j, and S_j \cap S_\ell = \varnothing for j \neq \ell. Consequently, for p \in \Delta_7, the outputs w_j(p) and w_\ell(p) have their "nontrivial" mass concentrated on disjoint coordinate sets. Since the b_j are supported on S_j (also disjoint), the images w_j(\Delta_7) and w_\ell(\Delta_7) are contained in disjoint open subsets of \Delta_7, giving w_j(\Delta_7) \cap w_\ell(\Delta_7) = \varnothing. The inclusion w_j(\Delta_7) \subset \Delta_7 is automatic. \square

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7. Explicit Example: Four Coarse-Graining Maps

7.1 Definition of the IFS

Set N = 8 and k = 4. Partition the coordinates into four disjoint pairs:

S_1 = \{1,2\}, \quad S_2 = \{3,4\}, \quad S_3 = \{5,6\}, \quad S_4 = \{7,8\}.

For each j \in \{1,2,3,4\}, the elementary stochastic matrix A_j moves one-half of the mass from coordinate 2j-1 to coordinate 2j, leaving all other coordinates unchanged. Concretely, A_j is the column-stochastic matrix defined by:

• Column 2j-1: \bigl(\frac{1}{2}, \frac{1}{2}, 0, \dots, 0\bigr)^\top distributed across rows 2j-1 and 2j;
• Column \ell for \ell \notin S_j: the standard basis vector e_\ell.

The action on p \in \Delta_7 is:

$$\begin{aligned} (A_j p){2j-1} &= \frac{p{2j-1}}{2}, \ (A_j p){2j} &= \frac{p{2j-1}}{2} + p_{2j}, \ (A_j p)\ell &= p\ell \quad \text{for } \ell \notin S_j. \end{aligned}

Each A_j is column-stochastic (columns sum to 1), and \sum_i (A_j p)_i = \sum_\ell p_\ell = 1.

For the offset, take b_j = \varepsilon \, \mathbf{1}_{S_j} where \varepsilon > 0 and \mathbf{1}_{S_j} is the indicator vector of S_j. Then B_j = 2\varepsilon and the normalized map is

w_j(p) = \frac{A_j p + \varepsilon \, \mathbf{1}_{S_j}}{1 + 2\varepsilon}.

7.2 Contraction factor

By Theorem G1, each w_j has Lipschitz constant

\lambda_j = \frac{1}{\sqrt{1 + 2\varepsilon}}.

Since all B_j = 2\varepsilon are equal, the global contraction factor is

\boxed{\lambda = \frac{1}{\sqrt{1 + 2\varepsilon}} < 1.}

Remark (connection to G3 — eigensolid fixed point). As \varepsilon \to 0^+, the maps approach the pure coarse-graining w_j(p) = A_j p, and \lambda \to 1. This reflects the fact that pure deterministic coarse-graining is only $1$-Lipschitz (not a strict contraction) in the Fisher metric, due to the existence of non-contracting directions (score functions constant on fibers). The \varepsilon > 0 regularization breaks this invariance and forces strict contraction.

The strand crossing operator C of Theorem G3 (see G3_EIGENSOLID_FIXED_POINT.md) is precisely this \varepsilon = 0 limit for the pair-averaging case: C equalizes each pair to their mean, is idempotent (C^2 = C), and is a non-strict contraction (\operatorname{Lip}(C) = 1). The maps w_j of G1 are thus the regularized (strictly contractive) version of G3's projection operator, with \varepsilon > 0 controlling the strictness. Conversely, G3's C is the degenerate limit of the G1 construction when the regularization is removed.

7.3 Explicit differential bound

For the map w_1 (moving half the mass from coordinate 1 to coordinate 2), with b_1 = (\varepsilon, \varepsilon, 0,\dots,0), the normalized image is

w_1(p)_1 = \frac{p_1/2 + \varepsilon}{1+2\varepsilon}, \quad w_1(p)_2 = \frac{p_1/2 + p_2 + \varepsilon}{1+2\varepsilon}, \quad w_1(p)_\ell = \frac{p_\ell}{1+2\varepsilon}\ \ (\ell \geq 3).

The differential is dw_1|_p(v) = A_1 v / (1+2\varepsilon), with (A_1 v)_1 = v_1/2 and (A_1 v)_2 = v_1/2 + v_2. Its squared Fisher norm is:

\|dw_1(v)\|_{F,w_1(p)}^2 = \frac{1}{(1+2\varepsilon)^2}\Biggl[\frac{v_1^2/4}{\bigl(p_1/2 + \varepsilon\bigr)/(1+2\varepsilon)} + \frac{(v_1/2+v_2)^2}{\bigl(p_1/2 + p_2 + \varepsilon\bigr)/(1+2\varepsilon)} + \sum_{\ell=3}^8 \frac{v_\ell^2}{p_\ell/(1+2\varepsilon)}\Biggr]

= \frac{1}{1+2\varepsilon}\Biggl[\frac{v_1^2/4}{p_1/2 + \varepsilon} + \frac{(v_1/2+v_2)^2}{p_1/2 + p_2 + \varepsilon} + \sum_{\ell=3}^8 \frac{v_\ell^2}{p_\ell}\Biggr].

Set u_\ell = v_\ell/p_\ell (the score). Then v_1 = p_1 u_1, v_2 = p_2 u_2, and the first two terms become:

\frac{p_1^2 u_1^2/4}{p_1/2 + \varepsilon} + \frac{(p_1 u_1/2 + p_2 u_2)^2}{p_1/2 + p_2 + \varepsilon}.

We claim this is bounded by (p_1 u_1^2 + p_2 u_2^2)/(1+2\varepsilon). For the first term:

\frac{p_1^2 u_1^2/4}{p_1/2 + \varepsilon} = p_1 u_1^2 \cdot \frac{p_1/4}{p_1/2 + \varepsilon} \leq p_1 u_1^2 \cdot \frac{1/4}{1/2 + \varepsilon} = \frac{p_1 u_1^2}{2(1+2\varepsilon)} \leq \frac{p_1 u_1^2}{1+2\varepsilon}.

For the second term, apply Jensen's inequality to the convex function t \mapsto t^2 with weights \alpha = (p_1/2)/(p_1/2 + p_2 + \varepsilon) and 1-\alpha:

\frac{(p_1 u_1/2 + p_2 u_2)^2}{p_1/2 + p_2 + \varepsilon} \leq \frac{p_1 u_1^2/2 + p_2 u_2^2}{1 + 2\varepsilon/(p_1 + 2p_2)} \cdot \frac{1}{1} \leq \frac{p_1 u_1^2 + p_2 u_2^2}{1+2\varepsilon}.

Adding the remaining terms \sum_{\ell=3}^8 v_\ell^2/(p_\ell(1+2\varepsilon)) gives

\|dw_1(v)\|_{F,w_1(p)}^2 \leq \frac{1}{1+2\varepsilon}\Bigl(p_1 u_1^2 + p_2 u_2^2 + \sum_{\ell=3}^8 p_\ell u_\ell^2\Bigr) = \frac{\|v\|_{F,p}^2}{1+2\varepsilon}.

Hence \kappa_1(p) \leq 1/\sqrt{1+2\varepsilon} for all p \in \Delta_7, confirming \lambda_1 = 1/\sqrt{1+2\varepsilon}. The same bound holds for w_2, w_3, w_4 by symmetry.

7.4 The attractor

With \lambda = 1/\sqrt{1+2\varepsilon}, the attractor \mathcal{A} \subset \Delta_7 has Hausdorff dimension (in the Fisher metric) satisfying

\dim_H(\mathcal{A}) \leq \frac{\log 4}{|\log \lambda|} = \frac{2\log 2}{\frac{1}{2}\log(1+2\varepsilon)} = \frac{4\log 2}{\log(1+2\varepsilon)}.

As \varepsilon \to 0^+, \dim_H(\mathcal{A}) \to \infty (the attractor expands to fill the simplex), while as \varepsilon \to \infty, \dim_H(\mathcal{A}) \to 0 (the attractor collapses to a finite set).

For \varepsilon = 1/2 (a natural choice since B_j = 1, matching the "1/2 mass" parameter), we obtain:

\boxed{\lambda = \frac{1}{\sqrt{2}} \approx 0.707.}

7.5 Summary of the example

Parameter	Value
Dimension N	8
Number of maps k	4
Fiber structure	\{1,2\}, \{3,4\}, \{5,6\}, \{7,8\}
Offset b_j	\varepsilon \cdot \mathbf{1}_{S_j}
Total offset B_j	2\varepsilon
Contraction factor \lambda	\displaystyle\frac{1}{\sqrt{1+2\varepsilon}}
For \varepsilon = 1/2	\lambda = 1/\sqrt{2}

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8. References

1. S. Amari, Differential-Geometrical Methods in Statistics, Springer Lecture Notes in Statistics 28, 1985.
2. M. F. Barnsley, Fractals Everywhere, Academic Press, 1988.
3. M. F. Barnsley and S. Demko, "Iterated function systems and the global construction of fractals," Proc. R. Soc. Lond. A 399 (1985), 243–275.
4. N. N. Chentsov, Statistical Decision Rules and Optimal Inference, Trans. Math. Monographs 53, AMS, 1982.
5. J. E. Hutchinson, "Fractals and self-similarity," Indiana Univ. Math. J. 30 (1981), 713–747.
6. A. Bhattacharyya, "On a measure of divergence between two statistical populations defined by their probability distributions," Bull. Calcutta Math. Soc. 35 (1943), 99–109.
7. C. R. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37 (1945), 81–91.

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End of proof.