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Explore circle dynamics and recent findings in circle maps with break or critical points in this comprehensive lecture. Gain insights into probability invariant measures, dynamical partitions, and hitting times of circle maps with singularities. Delivered by Akhtam Dzhalilov from Turin Polytechnic University in Tashkent, Uzbekistan, this 70-minute talk provides a concise introduction to the subject and delves into new developments in the field of mathematics.