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Explore the intricacies of smooth dynamics and symbolic dynamics in this hour-long conference talk delivered by Olivier Mathieu from Institut Camille Jordan. Presented as part of "The Pursuit of Symmetry: A Conference in Honour of the 80th Birthday of Robert V. Moody" at the Fields Institute, delve into the comparison and contrast between these two fundamental concepts in dynamical systems. Gain insights into the mathematical principles underlying both smooth and symbolic dynamics, and understand their applications and implications in the field of mathematics.