A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts

Let $A$ be an $N\times N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar {A}$ with entries in $\{0,1\}$ such that the associated Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_{\bar {A}}$ are isomorphic and $\det (1 -A) = - \det (1-\bar {A})$. As a consequence, we find that two Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, \sigma _A)$ is continuously orbit equivalent to either $(X_B, \sigma _B)$ or $(X_{\bar {B}}, \sigma _{\bar {B}})$.

Inhomogeneous random coverings of topological Markov shifts

Let $\mathscr{S}$ be an irreducible topological Markov shift, and let μ be a shift-invariant Gibbs measure on $\mathscr{S}$. Let (Xn)n ≥ 1 be a sequence of i.i.d. random variables with common law μ. In this paper, we focus on the size of the covering of $\mathscr{S}$ by the balls B(Xn, n−s). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim supn →+∞B(Xn, n−s) for every s ≥ 0, which depends on s and the multifractal features of μ. Our results include the inhomogeneous covering of $\mathbb{T}^d$ and Sierpinski carpets.

On the existence of maximizing measures for irreducible countable Markov shifts: a dynamical proof

We prove that if ${\Sigma }_{\mathbf{A} } ( \mathbb{N} )$ is an irreducible Markov shift space over $ \mathbb{N} $ and $f: {\Sigma }_{\mathbf{A} } ( \mathbb{N} )\rightarrow \mathbb{R} $ is coercive with bounded variation then there exists a maximizing probability measure for $f$, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case in the general irreducible non-compact setting. It is also noteworthy that our technique works for the full shift over positive real sequences.

Canonical compactifications for Markov shifts

We give a complete characterization of the compact metric dynamical systems that appear as boundaries of the canonical compactification of a locally compact countable state mixing Markov shift. Consider such a compact metric dynamical system. Then there is a pair of non-conjugate Markov shifts with conjugate canonical compactifications, one of which has the given compact system as canonical boundary.

On C*-Algebras Associated with Subshifts

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.