# Ergodic Theory meeting

#### Thursday 9th April

#### To be held in Room MS.01

##### Programme

1:30pm **Johannes Kautzsch** (Bremen)

*On the convergence to equilibrium of unbounded observables*

Abstract: We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $\beta \in [0, 1]$ and $\alpha \in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $\beta$ of order $\alpha$ is dependent on the structure of the $\omega$-limit set of $\beta$ with respect to $T_{r}$. This is joint work with M. Kesseböhmer and T. Samuel.

2:45pm** Julia Slipantschuk** (QMUL)

*Spectral structure of transfer operators for expanding circle maps*

Abstract: Spectral data of transfer operators yield insight into fine statistical properties of the underlying dynamical system, such as rates of mixing. In this talk, I will describe the spectral structure of transfer operators associated to analytic expanding circle maps. For this, I will first derive a natural representation of the respective adjoint operators. For expanding circle maps arising from finite Blaschke products, this representation takes a particularly convenient form, allowing to deduce the entire spectra of the corresponding transfer operators. These spectra are completely determined by the multipliers of attracting fixed points of the Blaschke products.

4:15pm** Benoit Saussol** (Brest)

*Poisson law for some non uniformly hyperbolic dynamical systems*

Abstract: We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. I will prove that the distribution of the number of visits to a ball B(x,r) converges to a Poisson distribution as the radius r -> 0 and after suitable normalisation.

On **Friday 10th April**, Johannes Kautzsch will give two further talks, both to be held in **MS.01**.

**Johannes Kautzsch**(Bremen)

*On the asymptotics of the $\alpha$-Farey transfer operator I*

**Johannes Kautzsch**(Bremen)

*On the asymptotics of the $\alpha$-Farey transfer operator II*

Abstract: We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $\alpha$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $\alpha$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point. This is joint work with M. Kesseböhmer, T. Samuel and B.O. Stratmann