Timeline for List of Bernoulli chaotic systems
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 1, 2015 at 18:02 | comment | added | Vaughn Climenhaga | Thanks for the clarification, Ian - I should have mentioned this. A full shift on an infinite alphabet has infinite topological entropy, but as Ian's comment makes clear, even in this case a fully-supported Bernoulli measure can have finite measure-theoretic entropy. | |
| Dec 1, 2015 at 17:04 | comment | added | Ian Morris | @demitau: there's no issue with a fully-supported Bernoulli measure on the countable full shift having finite entropy: the $(p_1,p_2,\ldots)$-Bernoulli measure on the full shift with countable alphabet has entropy $\sum_{n=1}^\infty -p_n\log p_n$, which is clearly finite if $p_n \ll n^{-1-\varepsilon}$. | |
| Dec 1, 2015 at 16:58 | vote | accept | demitau | ||
| Dec 1, 2015 at 16:54 | comment | added | Vaughn Climenhaga | Sarig's JMD paper considers a diffeomorphism $f$ of a surface $M$ with a positive entropy equilibrium state $\mu$ and shows that $(M,f,\mu)$ is Bernoulli; that is, the system itself is Bernoulli. He does this by obtaining a semi-conjugacy to $(M,f,\mu)$ from a countable-state shift. An infinite alphabet shift has infinite entropy if it is the full shift, but if it is just a Markov shift then it can have finite entropy (e.g., if not many transitions are allowed). Even if it is the full shift, it may have finite Gurevich pressure for certain potentials, which is what Sarig uses. | |
| Dec 1, 2015 at 16:52 | history | edited | Vaughn Climenhaga | CC BY-SA 3.0 |
Added links to Ledrappier's papers
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| Dec 1, 2015 at 16:42 | comment | added | demitau | About the Sarig results -- he proves that the system itself is Bernoulli (with countable alphabet), or that the corresponding induced system is Bernoulli? Infinite alphabet shifts should have infinite entropy, if I understand right and it puzzles me. For the flows there are also Sinai billiards and Lorenz flow, but seemingly Bernoullicity for a flow does not necessarily imply Bernoullicity for its time-1 map (or a Poincare map), although I don't see an immediate example showing why it is so. | |
| Dec 1, 2015 at 15:40 | history | answered | Vaughn Climenhaga | CC BY-SA 3.0 |