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Combining several results from the previous works of the author, three mainsubtleties will be clarified here. First, similar to the rotating BTZ blackholes, we show how for the warped BTZ black holes, the consideration of theeffective temperatures could solve the seemingly unsaturation issue of thechaos bound. Second, comparing the Hawking-Page phase diagrams of BTZ andwarped BTZ black holes in the topologically massive gravity and new massivegravity theories, we show how the characteristics of the action would specifythe behaviors of the chaos modes, and there we emphasize the importance ofusing the local and "physical" thermodynamical ensembles for studying theboundary chaos modes in warped CFTs and also the connections with the bulkreconstruction. Third, we propose that the boundary modular scrambling modeswhich saturate the modular chaos bound are related to the curvature invariantsin the geometrical side.

Chaos, Phase Transitions and Curvature Invariants of (rotating, warped, massive) BTZ Black Holes