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

This paper studies non-autonomous iterations of $\lambda e^z$, where at each step the parameter $\lambda$ 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 $1\mathrm{/}e<A<B<\mathrm{\infty }$. 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’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.