Timeline for A weak-mixing, zero entropy measure on the 2-shift which gives equal weight to both symbols
Current License: CC BY-SA 3.0
15 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Mar 9, 2016 at 15:53 | comment | added | Stéphane Laurent | Take the Chacon map and the stationnary process obtained by coding with a $(1/2,1/2)$-partition. It should work, no ? | |
| Mar 3, 2014 at 19:17 | history | edited | Ian Morris | CC BY-SA 3.0 |
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| Mar 3, 2014 at 18:20 | history | edited | Ian Morris | CC BY-SA 3.0 |
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| Mar 3, 2014 at 18:18 | vote | accept | Ian Morris | ||
| Mar 3, 2014 at 16:56 | answer | added | R W | timeline score: 5 | |
| Mar 3, 2014 at 15:22 | answer | added | Tom Kempton | timeline score: 3 | |
| Feb 28, 2014 at 14:27 | history | edited | Ian Morris | CC BY-SA 3.0 |
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| Feb 28, 2014 at 14:18 | comment | added | Ian Morris | Anthony: er, so it is! I had somehow miscalculated the determinant as being $2$ and got an eigenvalue of $\frac{5+\sqrt{17}}{2}$. Thank you! | |
| Feb 28, 2014 at 13:24 | comment | added | Anthony Quas | Ummm... the leading eigenvector of that matrix is (1|1) with eigenvalue 4: under the substitution, $01\mapsto 00111100$. | |
| Feb 28, 2014 at 10:49 | comment | added | Ian Morris | Dekking and Keane show that the substitution $0 \mapsto 001$, $1 \mapsto 11100$ is weak mixing. The measures of the cylinders $[0]$ and $[1]$ are determined by the limiting ratio of ones to zeros in the substitution, which in turn is given by the ratio of entries in the leading eigenvector of the matrix $(2 2 | 1 3)$. In particular I think that the measures of the cylinder sets are irrational for this substitution. I am a little concerned that irrational eigenvalues might be necessary for weak mixing but this is a good direction of inquiry I think. Thanks! | |
| Feb 28, 2014 at 9:31 | comment | added | Tom Kempton | Could you build such an example explicitly using substitutions? Take a substitution on symbols {0,1}, let \Sigma be the orbit closure of a periodic point for your substitution, the shift map on \Sigma has zero entropy and is uniquely ergodic. For the Thue Morse substitution, the invariant measure gives measure (1/2,1/2) to each symbol. Lots of people have studied conditions under which these systems are weak mixing (e.g. Dekking and Keane 83, which I don't have access to at home). Maybe you need three symbols to make weak mixing, but if there is an example here it's probably easy and explicit. | |
| Feb 27, 2014 at 17:55 | answer | added | Anthony Quas | timeline score: 4 | |
| Feb 27, 2014 at 17:44 | history | edited | Ian Morris |
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| Feb 27, 2014 at 17:32 | history | asked | Ian Morris | CC BY-SA 3.0 |