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No - because the definition of Harris recurrence contains an absolute continuity condition on transition probabilities. For an extreme counterexample just take any ergodic measure preserving transform …

This condition is much more restrictive than just having zero entropy. In fact you are talking about a purely atomic invariant measure of the shift on $A^{\mathbb Z_+}$, where $A$ is a finite alphabet …

What you call "duality" (that for two commuting actions under reasonable conditions ergodic properties of either action on the space of orbits of the other one are the same) goes back to the work of F …

It should follow from a general theorem of Guivarc'h on simplicity of the Lyapunov spectrum for products of matrices with Markov dependence http://www.ams.org/mathscinet-getitem?mr=772409

Ian's answer uses two two ingredients, which are not really necessary: the Birkhoff ergodic theorem and continuity of the state space. One can do it in a more direct way just in the Borel category. Le …

There are two edges going out of every vertex of this graph; label the vertical one with 0, and the other one with 1. Thus, paths in your graph can be parametrized by the space of 0,1 sequences. If on …

There is no continuity in the relationship between $h(X)$ and $\sigma^2$ either way. In one direction the example of Ian shows that the entropy can be arbitrarily close to $\log 2$, whereas $\sigma^2 …

The entropy of a flow $H=(h_t)$ is defined as the entropy of the time 1 transformation $h_1$. For any other $t$ the entropy of $h_t$ is $|t|$ times the entropy of $h_1$ (for example, see the book by C …

There is a more conceptual explanation. This is a direct corollary of the Markov property formulated in "invariant form": the past and and the future are conditionally independent with respect to the …

As far as |I understand your question, examples you want are provided by the so-called North-South dynamical systems, which are actions of $\mathbb Z$ (discrete time) or $\mathbb R$ (continuous time) …

Alas, there is a lot of confusion both in the answer (which is really misleading) and in the question (since the answer was accepted). First, about the original question of the author. If this is wha …

Instead of comparing the Poincare recurrence theorem with ergodic theorems one should rather look at the underlying notions of conservativity and ergodicity in the general context of a measure class p …

I presume that by an "invariant" map you mean one which is constant on the orbits of the action of $G$ on the configuration space. Therefore, it should factorize through the space of ergodic component …

First a general comment. Unfortunately, the categorical approach is completely missing in the way the measure theory is still taught nowadays, and the fact that there actually is only one "reasonable" …

The original work of Furstenberg "Non-commuting random products" (1963) actually contains an answer to your question in Theorem 8.6 which states the positivity of the top Lyapunov exponent for any ran …