Explore the dynamics of smooth chaotic surface diffeomorphisms in this plenary lecture from the 34th Brazilian Mathematics Colloquium (CBM). Delivered by Omri Sarig from the Weizmann Institute of Science, Israel, the talk delves into advanced mathematical concepts related to dynamical systems. Gain insights into this complex topic as part of the largest scientific gathering of the Brazilian mathematical community, held at IMPA in Rio de Janeiro from July 24-28, 2023. Learn from a renowned expert in the field and discover how this research contributes to the development and strengthening of mathematics in Brazil and beyond.