With the aid of estimates on widths of trajectory-horns and trajectory-harps from above, we show that the magnetic exponential map at an arbitrary point on a Hadamard Kähler manifold induces a homeomorphism of the unit tangent space onto the ideal boundary if the strength of magnetic field is not greater than the square root of the absolute value of the upper bound of sectional curvatures.
"Asymptotic Behaviors of Trajectories on a Hadamard Kähler Manifold." Tokyo J. Math. 43 (2) 481 - 495, December 2020. https://doi.org/10.3836/tjm/1502179311