# Abstracts & Slides

### Julien Cassaigne

Title: An aperiodic uniformly recurrent multidimensional word with low complexity
Abstract: We consider multi-dimensional infinite binary words (i.e., configurations in $$\{0,1\}^{\mathbb Z^d}$$). The complexity $$P(n)$$ of such a word is the number of different $n\times\cdots\times n$ patterns that occur in it. How small can $$P(n)$$ be when the word is aperiodic?
For $$d=1$$, we know (Morse and Hedlund theorem) that $$P(n) \geq n+1$$. For $$d=2$$, Nivat conjectured that $$P(n) \geq n^2+1$$, and it is known that $$P(n) \geq n^2/2$$ (Cyr & Kra 2015). However, for $$d\geq3$$, it does not hold that $$P(n) \geq n^3+1$$ (Sander & Tijdeman 2000).
In this talk we present a construction that shows that the complexity
of an aperiodic word in any dimension may be as low as $$O(n^2f(n))$$,
where $$f$$ is some function that tends to infinity, e.g. $$\log n$$.
Moreover, the word we construct is uniformly recurrent (i.e., it
generates a minimal subshift). We conjecture that it is not possible
to have $$P(n)=O(n^2)$$ when $$d\geq3$$.

### Paulina Cecchi

Title: Superlinear factor complexity and Invariant measures on subshifts
Abstract: Connections between the factor complexity of a subshift and its set of invariant measures have been widely studied since the work of Boshernitzan, who proved that a minimal subshift having nonsuperlinear factor complexity admits finitely many ergodic measures. Cyr and Kra showed that for any superlinear sequence $$(p_n)_{n\in \mathbb{N}}$$ there exists a minimal subshift $$(X,S)$$ having uncountably many ergodic measures and such that $$\liminf p_X(n)/p_n=0$$, where $$p_X$$ denotes the factor complexity function. In this talk I will present a generalization of this last result, which states that any Choquet simplex can be realized as the set of invariant measures of a Toeplitz subshift with arbitrarily low, superlinear factor complexity. This can also be seen as a generalization of a result by Downarowicz which says that Toeplitz subshifts can realize any Choquet simplex as their set of invariant measures. This is a joint work with Sebastián Donoso.

### Mark Piraino

Title: The Central Limit Theorem for Typical Cocylces
Abstract: We establish a central limit theorem for the maximal Lyapunov exponent of typical cocycles (in the sense of Bonatti and Viana) over irreducible subshifts of finite type with respect to the unique equilibrium state for a Hölder potential. We also establish other related results such as the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state and a large deviation principle. The transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Kiho Park.

### Godofredo Iommi

Title: Arithmetic and Geometric means in continued fractions
Abstract: In this talk I will discuss the size, measured by Hausdorff dimension, of sets of real numbers defined in terms of the arithmetic and geometric means of their continued fraction digits. This is joint work with Thomas Jordan.

### Samantha Fairchild

Title: Quadratic growth for pairs of saddle connections with bounded area
Abstract:
We will state a result motivated by counting special straight lines (aka saddle connections) on polygons in the plane with sides identified by translation (aka translation surfaces). A key component of this result uses the ergodicity of $$SL(2,R)$$ acting on the space of translation surfaces. We will state the result and the key dynamical components used in the proof of this theorem which are also used in previous results on counting saddle connections. This is joint work with Jayadev Athreya and Howard Masur.

### Dan Thompson

Title: Fluctuations of time averages around closed geodesics in non-positive curvature.

Abstract: We consider the Knieper-Bowen-Margulis measure of maximal entropy for the geodesic flow over a compact rank 1 non-positive curvature manifold. In joint work with Tianyu Wang, we show that certain approximations of this measure using regular closed geodesics asymptotically satisfy a type of Central Limit Theorem. This is the first statistical-type result that has been shown in this setting. However, we hope that it will not be the last – we will comment briefly on the history of this question and some possible future directions.