A local-to-global inequality for spectral invariants and an energy dichotomy for Floer trajectories

We study a local-to-global inequality for spectral invariants of Hamiltonians whose supports have a "large enough" disjoint tubular neighborhood on semipositive symplectic manifolds. As a corollary, we deduce this inequality for disjointly supported Hamiltonians that are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>C</mml:mi> <mml:mn>0</mml:mn></mml:msup> </mml:math> -small (when fixing the supports). In particular, we present the first examples of such an inequality when the Hamiltonians are not necessarily supported in domains with contact-type boundaries, or when the ambient manifold is irrational. This extends a series of previous works studying locality phenomena of spectral invariants [9, 13, 15, 20, 25, 27]. A main new tool is a lower bound, in the spirit of Sikorav, for the energy of Floer trajectories that cross the tubular neighborhood against the direction of the negative-gradient vector field.