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Results tagged with ergodic-theory
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user 8112
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
2
votes
Ergodic automorphism is mixing of all orders
Begin by thinking about the dual automorphism to a hyperbolic toral automorphism. You can project the dual group onto an "expansive" direction. Any relation of the sort you are looking at can be shift …
- 2,758
8
votes
Accepted
Ergodic theory and dynamical systems books references
For a beautiful overview, focusing on entropy and the variational principle, you can't beat Walters' book. I've given reading courses from it, and it is very well written and excellent for self-study. …
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6
votes
Furstenberg $\times 2 \times 3$ conjecture, bibliography
Klaus Schmidt in Vienna wrote (but never published) an extensive set of notes called "$\times$$\beta$, $\times$2, and $\times$3" (the $\beta$ refers to the $\beta$-transformation $x\to \beta x \pmod{1 …
6
votes
Accepted
Renewal systems: Intrinsic ergodicity and a question related to the Adler's conjecture
There are some examples related to your third question in "Renewal Systems, Sharp-Eyed Snakes, and Shifts of Finite Type" by Johnson and Madden, Amer. Math. Monthly 109 (2002), 258-272. A long time ag …
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10
votes
Accepted
Intuition of Kolmogorov-Sinai entropy
A good way to understand measurable entropy is via the Shannon-McMillan-Breiman Theorem. Roughly speaking it says that there is a constant $c$ so that most atoms $A$ in $\bigvee_{i=0}^{n-1} T^{-i}\mat …
- 2,758
2
votes
Extension of the definition of entropy to $\mathbb{Z}^d$ and $\mathbb{N}^d$
The extension of both topological and measure entropy to $\mathbb N^d$-actions is straightforward (in the case of measure entropy you just need to remember that you are taking inverse images of sets, …
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7
votes
Rate of convergence of ergodic averages related to irrational rotation
For a detailed account of this well-studied problem see, for example, Chapter 2 of Uniform Distribution of Sequences by L. Kuipers and H. Niederreiter.
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