Ergodic and dynamically non-recurrent escaping exponential map

Construct an escaping exponential map that is both ergodic and dynamically non-recurrent.

Background

The paper proves that every non-ergodic escaping exponential map is dynamically non-recurrent. It also discusses examples of escaping exponential maps that are both ergodic and dynamically recurrent. These results leave unresolved whether ergodicity can coexist with dynamical non-recurrence within the escaping exponential family.

The open problem therefore concerns the existence of an escaping parameter whose exponential map has ergodic Lebesgue dynamics while still admitting a positive-measure set disjoint from all of its positive iterates.

References

Thus the following interesting question remains open.

Is there an escaping exponential map which is ergodic but dynamically non-recurrent?

— Dynamical (non-)recurrence of escaping exponential maps  (2609.03610 - Cui et al., 3 Sep 2026) in Section 2, immediately following the theorem that every non-ergodic escaping exponential map is dynamically non-recurrent