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Karl Petersen (UNC-CH), Ergodic Theory and Dynamical Systems Seminar
November 8, 2016 @ 3:30 pm - 4:30 pm
Title: A well-known but still fascinating example in ergodic theory: The Gauss Map
Abstract: Defining Tx to be the fractional part of 1/x for x in the unit interval produces a map that is isomorphic to the shift map on continued fraction expansions. We review some interesting (long known) properties of this map. We will see that this map preserves a measure equivalent to Lebesgue measure, called the Gauss measure. The map is ergodic with respect to this measure, so using the Ergodic Theorem and properties of continued fractions one can determine average rate of growth of continued fraction denominators and digits and find that the typical speed of approximation of irrationals by rationals is the entropy of the map. We probably will not have time to discuss the related ideas of Farey (properly called Haros) fractions and denominators, the Minkowski ? function, and associated dynamical systems and C* algebras, mention of which can be found in the notes on the speaker’s web page.