Decay of Correlations in Random Quadratic Maps

Definitions and mathematical statement

Let $P$ be a probability measure on $[1,4]$ that is absolutely continuous with respect to Lebesgue measure on that interval. Let $(\lambda_i)_{i\geq 1}$ be independent, identically distributed random variables with common law $P$. The supplied random quadratic maps have the formula \[ T_\lambda(x)=\lambda x(1-x),\qquad \lambda\in[1,4]. \] For a parameter realization, the successive compositions are \[ T_{\lambda_n}\circ\cdots\circ T_{\lambda_1},\qquad n\geq 1, \] wherever the intended phase space makes these compositions a dynamical system.

The supplied question asks whether this system has random decay of correlations, described in the record as equivalent to being relatively exact. This is an incompletely specified question: the record does not specify the phase space for $x$, the reference or invariant measures, the observables and limiting expression defining random decay of correlations, or the measured dynamical system and factor relative to which exactness is intended. It also does not specify whether the assertion is to hold for every absolutely continuous $P$, or whether further restrictions on $P$ were present in the omitted source context. No rate of correlation decay is stated. Consequently, the asserted equivalence is preserved as source wording, but cannot be formulated or justified from the supplied data, and a complete quantified proposition cannot be recovered without additional source context.

Short English statement

The question concerns correlation decay and relative exactness for compositions of quadratic maps with independent parameters drawn from an absolutely continuous law on $[1,4]$. The supplied extraction omits the dynamical and measure-theoretic definitions needed for a precise proposition. Its open status and claimed equivalence are supplied source descriptions requiring review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001960$, title Decay of Correlations in Random Quadratic Maps, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems.
• Supplied report citation: Mini-Workshop: Thermodynamic Formalism, Geometry and Stochastics (2012), Oberwolfach Reports, pp. 77--129, DOI 10.4171/owr/2012/03. Source URL and supplied literature-source link labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/03. No individual proposer is identified in the record.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the extraction contains only the tail of the random quadratic map problem and does not state a complete proposition; it describes the requested general result as unresolved. This is supplied dated curation, not independently verified literature assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.
• The registry also supplies the statement-status label corrected_verified, with a claim that manual semantic, typographic, and LaTeX review found the cleanup self-contained and meaning-preserving. That supplied verification claim does not resolve the missing context identified by the supplied literature assessment.