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The strictly ergodic measures are dense in the ergodic measures in the $\bar d$-metric. The proof (actually, a sketch of the proof) is given in Single orbit dynamics by Weiss (Theorem 4.4', page 46).

I don't know wheter this counts as standard, but... There is a paper by Krystyna Ziemian: Rotation sets for subshifts of finite type. Fund. Math. 146 (1995), no. 2, pp. 189--201 containing some …

This is true, and follows from the property called coalescence. In a topological setting this definition is due to Auslander [Endomorphisms of minimal sets, Duke Math. J. Volume 30, Number 4 (1963), …

As Lee Mosher noted if $M$ is a compact metric space, then the shift map on the product space $X=M^\mathbb{N}$ has a dense orbit and hence it is transitive. Let me add a few remarks: There are two not …

Let $X$ be a strictly ergodic shift space, and $\omega_1$, $\omega_2$ be two different points in $X$. Is there an automorphism $\Psi$ of $X$ such that $\Psi(\omega_1)=\omega_2$? By an automorphism I m …

The proof of (2) is contained in the much more general theorem due to Sigmund. This is because the main result of Sigmund "Generic Properties Of Invariant Measures for Axiom A-Diffeomorphisms" Inventi …

Let $X$ be a compact metric space and $T\colon X\to X$ be a continuous map. The set of $T$-invariant Borel probability measures $\mathcal{M}_T(X)$ is well known to be non-empty, convex, compact, and m …

Maybe it is good to note that a similar question: What is the weak-$^*$ closure of the set $$ D=\{\mu\in\mathcal{M}_\sigma: \mu\text{ is isomorphic to some }\nu\in C\}? $$ has a dramatically different …

It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\n …

My question regards properties of the f-bar metric $\bar{f}$ defined for shift invariant measures on $\mathscr{A}^\infty$ where $\mathscr{A}$ is a finite alphabet. The definition of the $\bar{f}$ met …

A synchronized system is a transitive shift space $X$ which has a synchronizing block $v$, that is $v$ is an admissible block for $X$ and whenever $vw$ and $uv$ are admissible blocks in $X$, then $uvw …

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. …

A coded system [see F. Blanchard, G. Hansel, Systèmes codés, Theoretical Computer Science, Vol. 44, 1986, pp. 17-49, http://dx.doi.org/10.1016/0304-3975(86)90108-8. (http://www.sciencedirect.com/scien …

Does anybody know the reference to a proof of the following fact (which is not hard to prove, but seems to be well-known, see here): The space of shift-invariant measures under Ornstein's d-bar metric …

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, …