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##### The Relative Growth of Information

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**The Relative Growth of Information**Aimee S. A. Johnson Swarthmore College joint with Karma Dajani, Universiteit Utrecht Martijn de Vries, Technische Universiteit Delft**Given**. Scenario**Given**Express as decimal expansion . . Scenario**Given**Express as decimal expansion continued fraction expansion . . . Scenario**Given**Express as decimal expansion continued fraction expansion Question: Given first n digits in decimal exp, how many digits are known in continued fraction expansion? . . . Scenario**,**Rephrase: Given x starts with What is largest m=m(n,x) s.t. we know x starts with Lochs’ Theorem:**,**Rephrase: Given x starts with What is largest m=m(n,x) s.t. we know x starts with Lochs’ Theorem: for a.e. x,**Rephrase:**Given x starts with What is largest m=m(n,x) s.t. we know x starts with Lochs’ Theorem:**Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani,**Fieldsteel**Consider**Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Consider**Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Consider**Partition P Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Consider**PartitionP ._._._._._._._._._._. 0 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Consider**Or Partition P ._._._._._._._._._._. 0 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Consider**Or Partition P ._._._._._._._._._._. 0 .5 1 Partition Q .__._._.__.______. 0 .2 .25 .33 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel**Let**e.g. and Same for Question:**Tools**• Generating partition a.e. x≠y, there exists n s.t • Ergodic transformations • Entropy nonnegative number which indicates amount of uncertainty in system = hλ(S) • Shannon-McMillan-Breiman Theorem For T ergodic, P generating, a.e. x,**Theorem**Given 2 ergodic dynamical systems on [0,1) with generating interval partitions P and Q, with entropies c and d, for a.e. x,**Theorem**Given 2 ergodic dynamical systems on [0,1) with generating interval partitions P and Q, with entropies c and d, for a.e. x, e.g.**Higher Dimensions**• . • Assumptions • . • . • .**Theorem**• Given 2 ergodic dynamical systems on with generating partitions P and Q with entropies c and d, Then for a.e. x,**Vague idea of proof**Let M= When wouldn’t m(n,x) = M?**Vague idea of proof**Let M= When wouldn’t m(n,x) = M? bad points; where where x in is “frame” of**Vague idea of proof**Let M= When wouldn’t m(n,x) = M? bad points; where where x in is “frame” of So msr of bad pts ≤ msr of frames ≈ k**Vague idea of proof**Let M= When wouldn’t m(n,x) = M? bad points; where where x in is “frame” of So msr of bad pts ≤ msr of frames ≈ k So Σ(bad pts at nth stage) < So a.e. x leaves bad set eventually**With thanks to the organizers of the MSRI Connections for**Women, January 2007