Publications

2. Stationary measures associated to analytic one dimensional iterated function schemes (with Mark Pollicott). Mathematische Nachrichten (2018). (
__Wiley Online Library__
)

1. Entry time statistics to different shrinking sets, Stochastics and Dynamics. Stoch. Dyn., 17 (2017).
__ __
__(arXiv).__

-. The smoothness of the stationary measure.
__(arXiv)__.

-. A Large deviation and an escape rate result for special semi-flows. (
__arXiv__).

Abstracts

We study how the stationary measure associated to analytic contractions on the unit interval behaves under changes in the contractions and the weights. Firstly we give a simple proof of the fact that the integrals of analytic functions with respect to the stationary measure vary analytically if we perturb the contractions and the weights analytically. Secondly, we consider the special case of affine contractions and we prove a conjecture of J. Fraser in 3 on the Kantorovich–Wasserstein distance between two stationary measures associated to affine contractions on the unit interval with different rates of contraction.

We consider ψ-mixing dynamical systems (𝒳,T,ℬ,μ) and we find conditions on families of sets {𝒰n ⊂ 𝒳: n∈ℕ} so that μ(𝒰n)τn tends in law to an exponential random variable, where τn is the entry time to Un.

Let X be a compact metric space and let Ti, 1 ≤ i ≤ H be surjective continuous maps on X. We give necessary and sufficient conditions for the product of H smooth functions f1, f2, . . . , fH to be a continuous coboundary. As a consequence, we provide a topological analog of a result by I. Assani [arXiv:1805.07655].

This paper is devoted to study how do thermodynamic formalism quantities varies for time changes of suspension flows defined over countable Markov shifts. We prove that in general no quantity is preserved. We also make a topological description of the space of suspension flows according to certain thermodynamic quantities. For example, we show that the set of suspension flows defined over the full shift on a countable alphabet having finite entropy is open. Of independent interest might be a set of analytic tools we use to construct examples with prescribed thermodynamic behaviour.

We provide explicit formulaes for the first Kantorovich-Wasserstein distance between stationary measures for iterated function scheme on the unit interval. In particular, we consider two stationary measures with different configurations of the weights associated to the same iterated function schemes with disjoint images composed of: k positive contractions or 2 contractions of different sign. We also study the case of two stationary measures associated to different iterated function schemes.

We study the smoothness of the stationary measure with respect to smooth perturbations of the iterated function scheme and the weight functions that define it. Our main theorems relate the smoothness of the perturbation of: the iterated function scheme and the weight functions; to the smoothness of the perturbation of the stationary measure. The results depend on the smoothness of: the iterated function scheme and the weights functions; and the space on which the stationary measure acts as a linear operator. As a consequence we also obtain the smoothness of the Hausdorff dimension of the limit set and of the Hausdorff dimension of the stationary measure.

In this paper we consider a smooth flow (Λ,Φt) builded from suspending over a (non-invertible topologically mixing) subshift of finite type, and we equip it with an equilibrium measure ν on Λ. The two main theorems are a large deviation and an escape rate result. The first theorem gives an explicit formula for X>0 and Y such that

ν{x∈Λ:∣∫F∘Φs(x)ds−∫Fdμ∣>ϵ}≤exp(−Xt+logt+Y)

for t≫>1≫ϵ>0, where F:Λ→R is smooth. The second theorem gives an explicit lower bound for the asymptotic behaviour of the escape rate of ν through a small hole.

We prove an escape rate result for special semi-flows over non-invertible subshifts of finite type. Our proofs are based on a discretisation of the flow and an application of an escape rate result for conformal repellers.