On the Julia sets of the non-autonomous iteration of exponential maps

Abstract

This paper studies non-autonomous iterations of $λe^z$, where at each step the parameter $λ$ is chosen from a bounded interval. Urbański and Zdunik introduced and developed a thermodynamic formalism for random dynamics of exponential maps, extending the autonomous result established independently by Rees and Lyubich to typical parameter sequences in $[A, B]^{\mathbb{N}}$ with $1/e<A<B<\infty$. In this paper, we show that the same conclusion indeed holds for all parameters in $[A, B]^{\mathbb{N}}$, Our proof is based on carefully crafted uniform versions of pullback arguments, inspired by the original methods of Rees and Lyubich. In a related direction, Urbański and Zdunik established a non-autonomous analogue of Misiurewicz&#39;s theorem for the exponential map, later refined by Lech. We further extend their work by providing a refined condition for the Julia set to be the entire complex plane.