Explicit symbolic martingales for a sequential hyperbolic class

Definitions and mathematical statement

The source asks whether function spaces can provide martingale decompositions and concatenations for a class of hyperbolic maps. It leaves the class, measures, function spaces and meaning of concatenation unspecified. The attached aid treats a finite-state, time-inhomogeneous Markov coding as a restricted interpretation.

Let $X_n$ be a Markov chain on a finite set $E$, with transition kernels $P_n$, laws $\mu_{n+1}=\mu_nP_n$, and observables $f_n:E\to{\mathbb{R}}$ satisfying $\mu_nf_n=0$ and $\sup_n\|f_n\|_\infty<\infty$. Suppose the Dobrushin coefficients obey \[ \sup_n\frac12\max_{x,x'\in E}\sum_{y\in E}|P_n(x,y)-P_n(x',y)|\le\rho<1. \] On functions modulo constants this contracts the oscillation seminorm. The aid reports a uniformly bounded future Poisson corrector \[ h_n=\sum_{j=n}^{\infty}P_n\cdots P_{j-1}f_j, \qquad h_n=f_n+P_nh_{n+1}, \] where the empty product is the identity. With $\mathcal F_n=\sigma(X_0,\ldots,X_n)$, the increments \[ D_{n+1}=h_{n+1}(X_{n+1})-(P_nh_{n+1})(X_n) \] have conditional mean zero and give the exact forward decomposition \[ \sum_{n=0}^{N-1}f_n(X_n) =\sum_{n=0}^{N-1}D_{n+1}+h_0(X_0)-h_N(X_N). \] The endpoint is uniformly bounded, and truncating the future series gives an exponentially small lookahead error under the uniform contraction assumption.

For the explicit binary model $E=\{-1,1\}$, take \[ P_n(x,y)=\frac{1+a_nxy}{2},\qquad |a_n|\le\rho<1,\qquad f_n(x)=b_nx,\quad |b_n|\le B, \] with $X_0$ uniform, so all marginal laws remain uniform. Then the reported formulas specialize to \[ H_n=\sum_{j=n}^{\infty}b_j\prod_{k=n}^{j-1}a_k,\quad h_n(x)=H_nx,\quad D_{n+1}=H_{n+1}(X_{n+1}-a_nX_n). \] In particular $|H_n|\le B/(1-\rho)$, and the truncation at $j=n+K$ has error at most $B\rho^{K+1}/(1-\rho)$. The predictable variance is deterministic: \[ V_N=\sum_{n=0}^{N-1}{\mathbb{E}}(D_{n+1}^2\mid\mathcal F_n) =\sum_{n=0}^{N-1}H_{n+1}^2(1-a_n^2). \] The aid reports that $V_N\to\infty$ gives a standard Gaussian limit for $V_N^{-1/2}\sum_{n<N}b_nX_n$, using the bounded martingale increments and bounded endpoint. Loss of contraction at the boundary $a_n=1$ can invalidate the bounded-corrector argument; no conclusion for that boundary is inferred from the contracting case.

The aid also reports a moving-conjugate two-leg horseshoe realization as a sequential uniformly hyperbolic example. The supplied summary gives neither the conjugacies nor the invariant sets or coding measure. Consequently the symbolic formulas above are precise conditional statements; they are not a construction of that realization or a theorem for arbitrary concatenated hyperbolic maps or their physical measures.

Short English statement

Seek function-space methods for martingale decompositions along sequential hyperbolic dynamics. The supplied partial result uses uniformly contracting finite-state Markov codings, with an explicit binary corrector, variance formula and conditional CLT; a hyperbolic realization is reported without its construction.

Sources

• AIM workshop Stochastic methods for non-equilibrium dynamical systems, Problem 27: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001412, source code catalogue entry; canonical source aim-dynamical-systems-notes.json, notes[69].
• Attached machine-generated attempt 1: partial result, internal rigor label "proved", low novelty confidence; not peer reviewed and requiring independent verification before citation. No separate literature reference is supplied.