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1. The Relative Growth of Information Aimee S. A. Johnson Swarthmore College joint with Karma Dajani, Universiteit Utrecht Martijn de Vries, Technische Universiteit Delft

2. Given . Scenario

3. Given Express as decimal expansion . . Scenario

4. Given Express as decimal expansion continued fraction expansion . . . Scenario

5. 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

6. , Rephrase: Given x starts with What is largest m=m(n,x) s.t. we know x starts with Lochs’ Theorem:

7. , 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,

8. Rephrase: Given x starts with What is largest m=m(n,x) s.t. we know x starts with Lochs’ Theorem:

9. Consider Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

10. Consider Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

11. Consider Partition P Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

12. Consider PartitionP ._._._._._._._._._._. 0 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

13. Consider Or Partition P ._._._._._._._._._._. 0 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

14. Consider Or Partition P ._._._._._._._._._._. 0 .5 1 Partition Q .__._._.__.______. 0 .2 .25 .33 .5 1 Dynamical Systems Viewpoint Bosma, Dajani, Kraaikamp Dajani, Fieldsteel

15. Let e.g. and Same for Question:

16. 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,

17. 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,

18. 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.

19. Higher Dimensions • . • Assumptions • . • . • .

20. Theorem • Given 2 ergodic dynamical systems on with generating partitions P and Q with entropies c and d, Then for a.e. x,

21. Vague idea of proof Let M= When wouldn’t m(n,x) = M?

22. Vague idea of proof Let M= When wouldn’t m(n,x) = M? bad points; where where x in is “frame” of

23. 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

24. 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

25. With thanks to the organizers of the MSRI Connections for Women, January 2007