A DICHOTOMY FOR THE DIMENSION OF SOLENOIDAL ATTRACTORS ON HIGH DIMENSIONAL SPACE

The paper’s Main Theorem states the precise dichotomy for T(x,y)=(bx mod 1, γy+φ(x)) with φ real-analytic and 0<|γ|<1: either K is an analytic graph or, if ∆ is irrational, dim_H K = min{3, 1+log b/log(1/|γ|)}, with a rational-∆ lower bound; it also proves that the graph case occurs for at most countably many γ. The proof proceeds via symbolic coding S(x,j), the SRB measure ω=G(m×ν^ℕ), a gentle transversality condition (H) that holds unless K is a graph, Ledrappier–Young’s 1+α structure for dim(ω), a dimension-conservation property for fiber conditionals, and an inverse-entropy theorem in Rd; the rational case is reduced to a 1D-fiber problem by iterating T and projecting, then importing the real-fiber dichotomy. All these ingredients, including the countability argument for the graph case via analyticity in γ, appear explicitly in the paper’s Section 2 (expression/coding, (H) vs graph, countability), Theorem A (irrational ∆), and the rational reduction to [22]. The candidate solution follows the same structure and cites the same pillars (Tsujii coding, Ledrappier–Young, (H), Hochman inverse theorems, Ren’s real-fiber result), with a standard cohomological-equation criterion for the graph case. The only minor issue in the candidate’s outline is an imprecise derivation of a universal upper bound for dim_H K directly from dim(ω); the paper instead controls dim_B(K) ≤ min{3, 1+log b/log(1/|γ|)}, which together with dim(ω)=min{3, 1+log b/log(1/|γ|)} yields equality in the irrational case. Overall, both arguments agree in substance and conclusion, and the model’s additions (cohomological equivalence) are standard and compatible with the paper. See the Main Theorem statement and its proof sketch in Section 2, along with Theorem A, Theorem 3.1 (Ledrappier–Young), Theorem 4.1 (dimension conservation), and Theorem 5.1 (inverse theorem) in the paper .