Generalized Greatest Common Divisors for the Orbits under Rational Functions

Assume Vojta's Conjecture. Suppose $a, b, \alpha,\beta \in \mathbb{Z}$, and $f(x),g(x) \in \mathbb{Z}[x]$ are polynomials of degree $d \ge 2$. Assume that the sequence $(f^{\circ n}(a), g^{\circ n}(b))_n$ is generic and $\alpha,\beta$ are not exceptional for $f,g$ respectively, we prove that for each given $\varepsilon > 0$, there exists constant $C = C(\varepsilon,a,b,\alpha,\beta,f,g)>0$, such that for all $n \ge 1$, we have $$\gcd(f^{\circ n}(a)-\alpha, g^{\circ n}(b) -\beta) \le C\cdot\exp({\varepsilon\cdot d^n}).$$ We prove an estimate for rational functions and for a more general gcd and then obtain the above inequality as a consequence.