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Jet streams are narrow bands of strong westerly winds located near the tropopause and are therefore a fundamental part of extratropical atmospheric circulation. At the sub-seasonal scale, they are often studied using the weather regimes approach. In climate dynamics, weather regimes are loosely defined as a reduced set of large-scale, quasi-stationary, persistent, and recurrent flow patterns. Although weather regimes arise from different statistical classifications, the absence of a clear framework has weakened confidence in their practical application and even in their physical grounding. Most definitions rely on density considerations, associating a regime with a region of above-average density in phase space. A recent study proposes applying persistent homologies—a popular method in Topological Data Analysis—to weather regime detection from data, suggesting that the connected components of the highest density regions of a reconstructed phase-space point cloud correspond to weather regimes. However, results for the weather regime approach on the North Atlantic eddy-driven jet dynamics in winter yield only two of the three expected regimes. Here, we reconsider this question by focusing on the mechanisms acting in phase space to shape a flow in terms of recently developed mathematical concepts in the field of Chaos Topology. Our perspective provides a different interpretation of the results obtained for the North Atlantic region and is used to apply the weather regimes approach to the jet dynamics in winter over South America for the first time.