Zariski density of intersections of iterated subvarieties

Determine hypotheses on a variety over a field, an endomorphism of that variety, and subvarieties whose dimensions satisfy \(\dim V+\dim W\geq \dim X\), under which the union \(\bigcup_{i\geq 1}(\varphi^{n_i}(V)\cap W)\) is Zariski dense in \(W\) for every infinite subset \(\{n_i\}_{i\geq 1}\) of nonnegative integers.

Background

The paper proves a geometric likely-intersection theorem for product polynomial endomorphisms of P1×P1\mathbb{P}^1\times\mathbb{P}^1. Specifically, under ampleness and non-preperiodicity assumptions on one curve and a non-exceptionality assumption on the other, the union of intersections along any infinite sequence of iterates is Zariski dense.

The authors ask whether an analogous density principle holds for arbitrary varieties, endomorphisms, and subvarieties satisfying the expected dimension inequality. The question is presented as a generalization motivated by the paper's theorem and related results of Baldi and Urbanik.

References

Under what hypotheses can we conclude that $$\bigcup_{i \geq 1} \left({n_i}(V) \cap W \right) $$ is Zariski dense in $W$?

— Degree Growth of Iterates of Curves and Likely Intersections  (2609.20580 - Saleh et al., 17 Sep 2026) in Question 1, Introduction (labelled \ref{IntroQuestion1})

Under what hypotheses can we conclude that

\bigcup_{n=1}{\infty} (n(V) \cap W)

is Zariski dense in $W$?

— Degree Growth of Iterates of Curves and Likely Intersections  (2609.20580 - Saleh et al., 17 Sep 2026) in Question 2, Introduction (labelled \ref{weaker-question})