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The Relative Growth of Information

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

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The Relative Growth of Information

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

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

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

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

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

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

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

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

  17. 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,

  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,

  19. 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.

  20. Higher Dimensions • . • Assumptions • . • . • .

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

  22. Vague idea of proof .

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

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

  25. 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

  26. 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

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

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