Abstracts & Slides

Omri Sarig

Title: Effective intrinsic ergodicity for countable Markov shifts (joint with R. Ruhr)
Abstract: A theorem of Kadyrov says that for topologically transitive subshifts of finite type with measure of maximal entropy \(\mu_0\), for every shift invariant measure \(\mu\) and Hölder continuous function \(\psi\),

\(|\mu(\psi)-\mu_0(\psi)| \leq C |\psi|_\beta \sqrt{|h_\mu(T)-h_{\mu_0}(T)|},\)

where \(|\psi|_\beta\) is the Hölder norm of \(\psi\). We discuss to what extent this inequality extends to countable state Markov shifts.

Slides (PDF)

Anush Tserunyan

Title: Ergodic theorems along trees
Abstract: In the classical pointwise ergodic theorem for a probability measure preserving (pmp) transformation \(T\), one takes averages of a given integrable function over the intervals \(\{x, T(x), T^2(x),…., T^n(x)\}\) in the forward orbit of the point \(x\). In joint work with Jenna Zomback, we prove a “backward” ergodic theorem for a countable-to-one pmp \(T\), where the averages are taken over subtrees of the graph of T that are rooted at \(x\) and lie behind \(x\) (in the direction of \(T^{-1}\)). Surprisingly, this theorem yields (forward) ergodic theorems for countable groups, in particular, one for pmp actions of free groups of finite rank where the averages are taken along subtrees of the standard Cayley graph rooted at the identity. For free group actions, this strengthens the best known result in this vein due to Bufetov (2000).

Slides (PDF)

Wolfgang Steiner

Title: Matching for linear mod one transformations
Abstract: For a linear mod one transformation (also called intermediate \(\beta\)-transformation) \(T_{\beta,\alpha}(x) = \beta x + \alpha – \lfloor \beta x + \alpha\rfloor\), matching holds if \(T_{\beta,\alpha}^n(0) = T_{\beta,\alpha}^n(1^-)\) for some \(n\). In this case, the absolutely continuous invariant measure has piecewise constant density. Bruin, Carminati and Kalle (2017) proved that matching occurs for almost all \(\alpha\) when the base $\beta$ is a quadratic Pisot number or the Tribonacci number, and they conjecture that this holds for all Pisot numbers. We discuss for which bases matching can occur, show that \(0\) is an accumulation point of matching parameters when \(\beta\) is a Pisot number or a simple Parry number, and address relations with intermediate \(\beta\)-shifts of finite type. We also discuss matching properties for \(\alpha\)-continued fractions.

Slides (PDF)

Jon Fickenscher

Title: Bounding Partial Rigidity in Rank-One Systems
Abstract: Rank-One Transformations may be defined by cutting parameters \((q_n)\) and spacer parameters \((s_{n,i})\). For a given Rank-One Transformation, S. Gao and A. Hill defined the canonical sequences of these parameters and showed that, when these sequences are bounded, the transformation has trivial centralizer. Such a transformation therefore cannot be rigid.

For a probability system \((X,T,\mu)\), we call a sequence \((n_m)\) an \(\alpha’\)-rigidty sequence if \(\liminf_{n\to\infty} \mu(A \cap T^{-n_m} A) \geq \alpha’ \cdot \mu(A)\) for all measurable \(A\subset X\). We say a system has partial rigidity constant \(\alpha\) if \(\alpha\) is the supremum of all such \(\alpha’\). By this definition, rigidity is equivalent to \(\alpha = 1\). We bound \(\alpha\) away from one for a class of Rank-One Transformations that are canonically bounded and investigate rigidity in unbounded cases.

This is ongoing work with Kelly Yancey.

Slides (PDF)

José Rigoberto Zelada

Title: Strongly mixing actions of countable abelian groups are almost strongly mixing of all orders
Abstract:
Let \((G,+)\) be a countable discrete abelian group and let \((T_g)_{g\in G}\) be a strongly mixing measure preserving \(G\)-action on a probability space \((X,\mathcal A,\mu)\). We will present a result which states that \((T_g)_{g\in G}\) is “almost strongly mixing of all orders”. The proof relies on a new notion of largeness for subsets of \(G^d\) and utilizes \(\mathcal R\)-limits, a notion of convergence based on the classical Ramsey Theorem. This talk is based on joint work with Dr. Vitaly Bergelson.

Slides (PDF)

Benjamin Weiss

Title: Generic behavior in positive entropy

Abstract: In September 2003, Dan Rudolph gave a talk at the dynamics seminar at the Univ. of Maryland with the above title. The surprising result that he discussed in that talk was the following. If 0 < c < log 2 and \(M_c\) is the space of shift invariant measures on \(\{0,1\}^Z\) such that their entropy is at least c then for for a generic set of measures in \(M_c\) the resulting system is isomorphic to a Bernoulli shift with entropy c. I will discuss this result and some generalizations. (Joint work with Jean-Paul Thouvenot)

Slides (PDF)