Finite random iterated function systems do not always satisfy Bowen's formula
Yuya Arima
TL;DR
The paper addresses whether Bowen's formula for random iterated function systems extends to finite RIFS. It constructs a finite RIFS with a frame and a tail distribution $p_n=1/(C n^2)$ that yields almost surely $\dim_H(J(\Psi(\omega)))=0$ while the Bowen parameter satisfies $B(\Psi)=d$. The method combines a nonautonomous conformal IFS construction, frame-based indexing, and ergodic/Khinchin-type arguments to bound the random pressure and derive a strict inequality. The result demonstrates a limitation in transferring Bowen-type dimension formulas from random recursive constructions to RIFS, highlighting the need for additional structural hypotheses.
Abstract
In this paper, we provide a finite random iterated function system satisfying the open set condition, for which the random version of Bowen's formula fails to hold. This counterexample shows that analogous results established for random recursive constructions are not always obtained for random iterated function systems.