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You can't do substantially better than $\|f_k\|_\lambda$. Here's a simple example: Consider $X=\{0,1,\ldots,2^{N}-1\}$, equipped with normalized counting measure. The transformation is $T(x)=x+1\bmod …

No. I don't think there can be any result of that type. Let $X=\{0,1\}^{\mathbb N}$ and let $S$ be the dyadic odometer (I like to write elements of $X$ so they are infinite on the left; in this case, …

No. For example, take $S=\mathbb R$ and let $\mu_n$ be Lebesgue measure restricted to $[n,n+1]$. Set $$ f(x_0,x_1,\ldots)=\sin^2(2\pi x_0)\sin(2\pi(x_{2^j-\lfloor x_0\rfloor}) \text{ if $x_0\in [2^{j …

There are counterexamples. The easiest way to “cheat” is to let the height function be cohomologous to a constant. As an example, let $T$ be an ergodic transformation of a space $X$. Let $A$ be a subs …

So your question can be rephrased this way: $\theta$ maps $\Omega$ to itself and preserves the measure $\mathbb P_W$. You are then asking about a skew product extension of $\theta$: $\bar\theta:\Omega …

So I think there's a very direct argument to show that if $\mathbb P_W\times\lambda$ is ergodic then it's the unique invariant measure projecting onto $\mathbb P_W$. Let $\bar\Omega=\Omega\times S^1$ …

I think you're exactly asking for the strong mixing dynamical systems with singular spectral type. One comment: these must have entropy 0 as any positive entropy measure has Lebesgue spectrum.

$T\times R_\theta$ is ergodic if and only if $T$ is ergodic and $e^{2\pi im\theta}$ is not an eigenvalue of $T$ for any $m\in \mathbb Z\setminus\{0\}$. For an idea of the proof, let $T\colon X\to X$. …

Quite a nice one is the two-set partition $A,A^c$, where $A$ is the set of points with an even number of terminal 0's: $$ A = \bigcup_{k \geq 0} A_k $$ where $A_k = \bigl\{(x_1, x_2, \ldots) \mid x_1 …

If you're dealing with a probability space, the constant function $\mathbf 1$ belongs to $L^1$, so that you have no choice but to normalize by $n$ if you want to take Cesàro averages of $L^1$ function …

You're right. The set of measures for these maps is a zoo! There is the obvious map (base $d$ expansion) $\pi$ from $\{0,1\ldots,d-1\}^{\mathbb N}$ to $[0,1)$ which is a bijection off a countable set. …

What you are asking for is a conjugacy of the dynamical systems $g(x)=4x(1-x)$ and $h(x)=\sin(\pi x)$. Since $g$ and $h$ are both full unimodal maps of $[0,1]$, there will exist such a conjugacy, but …

I don't think your condition implies anything much. Let $(\epsilon_t)$ be your favourite ergodic 0--1 valued process; and let $X_t=\sum_{j=1}^\infty 2^{-j}\epsilon_j$. Then the function $\phi$ is $\ph …

As you say, it's an expanding map (min derivative at 0 is $\log 3$), so Lasota-Yorke and a bunch of other papers give that it has an absolutely continuous invariant measure. It's too much to hope that …

I think there is a counter-example. We'll build a subshift on 2 symbols, 0 and 1. Given a word $W$ with symbols 0 and 1, $\overline W$ will denote the word with all symbols flipped. Let $W_0=1$ and …