## ergodic theory – Number of permitted words up to permutation in a subshift

Let $$A$$ be a finite set and let $$X subseteq A^{mathbb{N}}$$ be a subshift. Let $$mathcal{L}_n$$ denote the set of words of length $$n$$ appearing in $$X$$. For a word $$w in mathcal{L}_n$$, one can define its empirical distribution to be the probability vector $$P_w := left( frac{N(a | w)}{n} right)_{a in A}$$ where $$N(a | w)$$ denotes the number of occurrences of the letter $$a in A$$ in the word $$w$$. I am interested in the quantity $$T(n) := # { P_w : w in mathcal{L}_n }$$ (equivalently, $$T(n) = #(mathcal{L}_n / sim)$$, where $$sim$$ is the equivalence relation on $$mathcal{L}_n$$ that declares two words to be equivalent if they are permutations of each other).

More specifically, I’m interested in subshifts with the property that $$limsup_{n to infty} T(n) = infty$$. Is there a name for this property? Are there any more commonly known conditions that imply this property?

Some examples/non-examples for intuition:

1. $$X = {0,1}^mathbb{N}$$. All words are permitted, so $$T(n) = n+1$$.

2. $$X subseteq {0,1}^mathbb{N}$$ is the SFT defined by forbidding the word $$11$$. Words of length $$n$$ can have up to half of their entries be equal to $$1$$, so $$T(n) approx n/2$$.

3. $$X$$ is the orbit closure of $${ x in {0,1}^mathbb{N} : x_{2i}x_{2i+1} = 01 text{ or } 10 text{ for all } i }$$. Any permitted word $$w$$ of length $$n$$ has both $$N(0 | w), N(1 | w) in (n/2 – 2, n/2 + 2)$$, so $$T(n) leq 5$$ for all $$n$$.

## markov chains – Argument in proof of ergodic theorem in Bremauld

i am stuck again in the proof of the ergodic theorem of Bremauld’s book “Markov Chains and Gibbs Measures, 2nd edition” on page 130 in the proof of proposition 3.3.1.

The equation at question is

$$frac{sum_{k=1}^{tau_{v(n)}}f(X_k)}{v(n)}leq frac{sum_{k=1}^nf(X_k)}{v(n)}leq frac{sum_{k=1}^{tau_{v(n)+1}}f(X_k)}{v(n)}$$

We know at that point that

$$frac1nsum_{k=1}^{tau_{v(n)}}f(X_k)xrightarrow{f.s.}sum_{iin E}f(i)x_i$$

Regarding that $$v(n)=sum_{k=1}^nmathbf 1_{{X_k=0}}$$
I don’t get how “the extreme terms of the above chain of inequalities tend to $$sum_{iin E}f(i)x_i$$ as $$nrightarrowinfty$$” shows the convergence…

Can anyone help?

## Probability – A subadditive maximum ergodic theorem

To let $$( Omega, mathcal A, operatorname P)$$ be a probability space $$tau: Omega to Omega$$ a measurable card on it $$( Omega, mathcal A)$$ With $$operatorname P circ : tau ^ {- 1} = operatorname P$$, $$Y_n: Omega to (- infty, infty)$$ Be $$mathcal E$$-Measurable for $$n in mathbb N$$ With $$operatorname E left (Y_1 ^ + right) < infty$$ and $$Y_ {m + n} le Y_m + Y_n circ tau ^ m ; ; ; text {for all} m, n in mathbb N tag1$$ and $$M_n: = max (Y_1, ldots, Y_n) ; ; ; text {for} n in mathbb N.$$

It is easy to show the following extension of the maximum ergodic theorem: $$operatorname E (Y_1; M_n ge0) ge0 ; ; ; text {for everyone} n in mathbb N. tag2$$

The usual maximum ergodic theorem is given by the special case, whereby $$Y_n = sum_ {i = 0} ^ {n-1} X circ tau ^ i ; ; ; text {for everyone} n in mathbb N$$ for an integrable real random variable $$( Omega, mathcal A, operatorname P)$$. In this special case it can be derived $$(2)$$ The $$operatorname P left ( sup_ {n in mathbb N} left | frac {Y_n} n right | ge c right) le frac1c operatorname E (| Y_1 |) ; ; ; text {for everyone} c> 0 tag3.$$

Can we extend this result to the general case?

## Probability – entropy rate problem of the ergodic Markov process with non-ergodic joint

I have a problem with the entropy rate when two independent Ergodic Markov processes have a non-ergodic joint. Let us take a closer look at two finite Markov processes $$mathscr {P} _1$$ and $$mathscr {P} _2$$ with transition matrices $$Pi_1$$ and $$Pi_2$$, respectively. Let’s assume that $$Pi_i$$. $$i = 1, 2$$is an irreducible line-stochastic period matrix $$p_i$$, and $$gcd (p_1, p_2) = p> 1$$, because $$Pi_i$$. $$i = 1, 2$$is irreducible, we know that $$mathscr {P} _i$$ is ergodic.

Assume that $$mathscr {P} _1$$ and $$mathscr {P} _2$$ are independent, we have their common process has transition matrix $$Pi = Pi_1 otimes Pi_2$$, Where $$otimes$$ means the Kronecker product. Since then, however $$p> 1$$. $$Pi$$ is Not irreducible. In fact, we can rearrange the columns and rows $$Pi$$ at the same time so that it will $$textrm {diag} (A_1, ldots, A_p)$$ Where $$A_i$$. $$i = 1, dots, p$$is an irreducible line-stochastic matrix.

My question is: We denote the entropy rate of the Markov process, which is given by the transition matrix $$A$$ by $$mathcal {H} (A)$$, do we have $$mathcal {H} (A_i) = mathcal {H} ( Pi_1) + mathcal {H} ( Pi_2)$$ for all $$i = 1, dots, p$$?

The statement applies to some examples that I tried (some of which are not even Markov processes, but were generated by somewhat more complicated models). However, I was unable to provide evidence or proof of disproval.

## Invariant Ergodic Measure Volterra Operator

Define the Volterra operator $$V$$ on $$C_0 ((0,1)) triangleq {g in C ((0,1)): g (0) = 0 }$$ by
$$f mapsto int_0 ^ { cdot} f (s) ds.$$
Is there an example of an ergodic and $$V$$Borel probability measure on $$C_0 ((0,1))$$?

## For the invertibility of ergodic averages!

To let $$x$$ Reversible, unlimited operator $$mathrm {II_ {1}}$$ factor $$(M, tau)$$, Under which condition? $$x$$iterates too $$1 + sigma (x) + cdots + sigma ^ {n} (x)$$ are reversible for everyone $$n$$ $$in$$ $$mathbb {N}$$? Where, $$sigma$$ $$in$$ $$Aut (M)$$? Inversity here means that the kernel is trivial.

## first ergodic transformation

An invertible ergodic transformation $$sigma: ( Omega, mathbb {P}) to ( Omega, mathbb {P})$$,

Given a random function $$C: Omega to mathbb {R} ^ {+}$$ s.t. Heck estimate $$mathbb {P} (C> k) le frac {1} {k ^ 2}$$,

Define a first meeting time $$tau ( omega) = inf {k: C ( sigma ^ k omega) le L }$$ with a fixed size $$L> 0$$,

we can show $$int tau ^ 2 d mathbb {P} < infty$$ for some big ones $$L$$? or has counter example? Many Thanks!

## sp.spektraltheorie – correlation between convergence rates in ergodic theorems and the property of the spectral gap

I read Quantitative ergodic theorems and their number-theoretic applications From Gorodnik and Nevo (arXiv: 1304.6847). At the beginning, there is a commentary on the convergence rates in the middle ergodic theorem that confuses me.

In short, leave $$G$$ be a (locally compact second countable) group that acts on a measuring space $$X$$, To let $$F_t subset G$$, and let $$beta_t$$ denote the uniform probability measure supported by $$F_t$$, To let $$pi_X ( beta_t)$$ designate the operator $$L ^ 2 (X)$$ given by $$pi_X ( beta_t) f: = | F_t | ^ {- 1} int_ {F_t} f (g ^ {- 1} cdot x) dg$$, On p. 8, the authors note that for a properly ergodic action of a countable accessible group $$G$$ on $$X$$that holds it $$Vert pi_X ( beta_t) Vert = 1$$,

I believe that. But right after that, they say that "for a truly ergodic action of a countable accessible group, no uniform convergence rate on average, the ergodic theorem can be established, namely the norm of averaging
Operators do not disintegrate at all. "(Highlighting her)

But, is this really the truth? In other words, is the (well-known) lack of a uniform convergence rate of the mean ergodic theorem for accessible groups actually only a direct consequence of the above-mentioned fact about the norms of the average operators?

## Ergodic theory – measures to complete a group in terms of payable many subgroups

To let $$G$$ to be a group and

$$G_0 subset G_1 subset ldots subset G_n subset ldots subset G$$
countable many subgroups.

To let $$Gamma: = varprojlim_ {n ge 0} G / G_n$$,

does $$Gamma$$ to have a $$G$$-Invariant measurement that meets the characteristics of hair measurement in a profinite group?

Note that $$Gamma$$ is generally not a topological group (or even a group) because we do not accept any of them $$G_n$$ to be normal