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Metaverse has recently attracted much attention from both academia and industry. Virtual services, ranging from virtual driver training to online route optimization for smart goods delivery, are emerging in the Metaverse. To make the human experience of virtual life more real, digital twins (DTs), namely digital replicas of physical objects, are key enablers. However, DT status may not always accurately reflect that of its real-world twin because the latter may be subject to changes with time. As such, it is necessary to synchronize a DT with its physical counterpart to ensure that its status is accurate for virtual businesses in the Metaverse. In this paper, we propose a dynamic hierarchical framework in which a group of IoT devices is incentivized to sense and collect physical objects' status information collectively so as to assists virtual service providers (VSPs) in synchronizing DTs. Based on the collected sensing data and the value decay rate of the DTs, the VSPs can determine synchronization intensities to maximize their payoffs. In our proposed dynamic hierarchical framework, the lower-level evolutionary game captures the VSPs selection by the IoT device population, and the upper-level differential game captures the VSPs payoffs, which are affected by the synchronization strategy, IoT devices selections, and the DTs value status, given VSPs are simultaneous decision makers. We further consider the case in which some VSPs are first movers and extend it as a Stackelberg differential game. We theoretically and experimentally show that the equilibrium to the lower-level game exists and is evolutionarily robust, and provide a sensitivity analysis with respect to various system parameters. Experiments show that the proposed dynamic hierarchical game outperform the baseline.