# Abstracts

ABSTRACTS

**Jon Aaronson** (Tel Aviv)

Title: *Absolute normalization of Birkhoff sums in infinite ergodic theory*

Abstract: Let $T$ be a conservative, ergodic measure preserving transformation of a $\sigma$-finite, infinite measure space. Pointwise, asymptotic normalization of one-sided Birkhoff sums fails: for every sequence of constants $a_n >0$ and $f \in L^1_+$, $\sum_{k=0}^n f\circ T^k \not\asymp a_n$ a.e. Recent examples (of Maucourant-Schapira) show that there could be constants $a_n>0$ so that $\sum_{k=-n}^n f\circ T^k \asymp a_n$ a.e. $\forall \ f \in L^1_+$. Nevertheless, there is never a.e. convergence of absolutely normalized, symmetric Birkhoff sums. I'll discuss this and alsi some other examples of these phenomena. This is joint work with Benjamin Weiss and Zemer Kosloff.

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Title: *Small bases that admit countably many expansions*

Abstract: Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\epsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base~$q$ (or a $q$-expansion) if

$$x=\sum_{i=1}^{\infty}\epsilon_iq^{-i}.$$

Let $\mathcal{B}_{\aleph_{0}}$ denote the set of $q$ for which there exists $x$ with exactly $\aleph_{0}$ expansions in base $q$. It is a well known result that $\min\mathcal{B}_{\aleph_{0}}=\frac{1+\sqrt{5}}{2}.$ In this talk we discuss a recent result which states that the smallest element of $\mathcal{B}_{\aleph_{0}}$ strictly greater than $\frac{1+\sqrt{5}}{2}$ is $q_{\aleph_{0}}\approx1.64541$, the appropriate root of $x^6=x^4+x^3+2x^2+x+1$.

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Title: *Equivalence classes of cut-and-project nets*

Abstract: Separated nets (a.k.a. Delone sets) can be constructed from R^{d} actions on the torus, via the cut-and-project method. This method produces examples of aperiodic patterns which occur in nature (e.g. in quasicrystals), and it is desirable to understand how 'close’ these patterns are to periodic. In previous joint work we showed that generically, cut-and-project nets are bi-Lipschitz equivalent to a lattice, and that, for some choices of dimensions, they are generically bounded distance to a lattice. The goal of this talk is to explain a recent proof that, in any co-dimension one cut-and-project setup, regardless of Diophantine properties, the acceptance domain can always be chosen in a non-trivial way so that the resulting separated net is bounded distance to a lattice.

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**Oliver Jenkinson** (QMUL)

Title: *Joint spectral radius and ergodic optimization*

Abstract: We construct new counterexamples to the Lagarias-Wang finiteness conjecture on the joint spectral radius. The approach uses ergodic optimization and Sturmian measures. This is joint work with Mark Pollicott.

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Title: *Subword complexity and Sturmian colorings of regular trees*

Abstract: In this talk, we introduce subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and then introduce Sturmian colorings, which are colorings of minimal unbounded subword complexity.We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ''infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones. We will provide several examples. This is joint work with Seonhee Lim.

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**Zemer Kosloff **(Warwick)

Title: *A class of inhomogeneous Markov shifts without an absolutely continuous invariant measure*

Abstract: An inhomogeneous Markov shift is the shift of a Markov chain where the transitions may vary over time. Topological Markov shifts are a class of symbolic topological dynamical systems which play a crucial role in the study of dynamical systems arising from differential equations. In this talk we will discuss the construction of examples of Markov shifts supported on a topological Markov shift which have no absolutely continuous invariant $\sigma$-finite measure.

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**Ian Melbourne** (Warwick)

Title: *An operator renewal equation for continuous time dynamical systems*

Abstract: We introduce an operator renewal equation for dynamical systems with continuous time. Using this equation, we prove results on mixing and rates of mixing for both finite and infinite measure flows. This is joint work with Dalia Terhesiu.

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**Sara Munday** (York)

Title: *Diophantine approximation and colouring*

Abstract: The goal of this talk is to bring to light some recently discovered connections between problems about graph colourings and problems about the approximation of real numbers by rationals. The connections arise as a result of the fact that many statements about the quality of approximation of real numbers can be phrased as problems about the orbits of points in certain spaces (e.g. compact metric spaces) under the action of particular groups. Once these group actions are identified, there is a correspondence between questions about the approximations and questions about the Cayley graph of the given group. Information on either side of this correspondence gives information about the other. Several authors have used this machinery to transfer information from Diophantine approximation to give upper bounds for the chromatic number of Cayley graphs. Our main result shows that interesting information about Diophantine approximation can also be obtained by going in the other direction.We show how lower bounds for the chromatic number of certain Cayley graphs can be used to give a new proof of the p-adic Littlewood Conjecture for quadratic irrationals.

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**Hitoshi Nakada** (Keio)

Title: *On the duality between piecewise rotations of the circle and interval exchange maps*

Abstract: We construct a family of piecewise rotations of the circle from the natural extension of the Rauzy induction. This shows a sort of duality between interval exchange maps and piecewise rotation of the circle.

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**Richard Sharp **(Warwick)

Title: Convergence of orbital measures for geodesic and hyperbolic flows

Abstract: A theorem of Bowen (1972) asserts that the periodic orbits of a hyperbolic flow, with periods in an interval [T-a,T+a], become

equidistributed with respect to the measure of maximal entropy, as T tends to infinity. For geodesic flows over manifolds of constant negative curvature, this measure agrees with the Liouville measure. In variable negative curvature, the periodic orbits are no longer equidistributed with respect to Liouville measure but, by a result of Parry (1986), an appropriately weighted average is and this result extends to hyperbolic attactors and SRB measures. We will discuss results where the interval is allowed to shrink in this setting. This is joint work with Mark Pollicott.

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**Nikita Sidorov** (Warwick)

Title: *Open maps: from 1D to 2D*

Abstract: Let T be a map on a compact metric space X and let H be an open subset of X. Let J(H) denote the complement of the union of all T-preimages of H. The restriction of T to J(H) is called an open map. In this talk my main concern will be the size of J(H) - more specifically, whether this set is uncountable or not. A complete analysis will be presented for the case of the doubling map and an interval H, whereas for the baker's map I will present some preliminary findings which indicate major differences with the one-dimensional case. This talk is based on my recent joint paper with P. Glendinning as well as a work in progress joint with L. Clark.

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**Hiroki Takahashi** (Keio)

Title: *Lyapunov exponents of ergodic measures at the first bifurcation of the H\'enon family*

Abstract: We consider the dynamics of the strongly dissipative Hénon map around the first bifurcation parameter $a^*$ at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. In [Takahasi, H.: Commun. Math. Phys. 312, 37-85 (2012)], it was proved that $a^*$ is a full Lebesgue density point of the set of parameters for which Lebesgue almost every initial point diverges to infinity under forward iteration. For these parameters, we show that all Lyapunov exponents of all ergodic Borel probability measures are uniformly

bounded away from zero, uniformly over all the parameters.

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