Related papers: On Gauss-Kuzmin Statistics and the Transfer Operat…
We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these…
We consider a family of continued fraction expansions of any number in the unit closed interval $[0,1]$ whose digits are differences of consecutive non-positive integer powers of an integer $m \geq 2$. For this expansion, we apply the…
Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable…
We consider a family of operators connected with the geodesic flow on the modular surface. We show certain spectral information is retained after expanding their domain to the space of $\alpha$-H\"older continuous functions on the unit…
The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->{1/x} eventually reach…
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the…
Graphs and networks play an important role in modeling and analyzing complex interconnected systems such as transportation networks, integrated circuits, power grids, citation graphs, and biological and artificial neural networks. Graph…
We consider an interval map which is a generalization of the R\'enyi transformation. For the continued fraction expansion arising from this transformation, we prove a result concerning the asymptotic behavior of the distribution functions…
In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the…
We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator $q$, approaches the Gauss-Kuzmin statistics with polynomial rate in $q$. This improves on…
The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint…
Transfer operators have been used widely to study the long time properties of chaotic maps or flows. We describe quantum analogues of these operators, which have been used as toy models by the quantum chaos community, but are also relevant…
Let $(X,d)$ be a metric space and $X_0$ be an open and dense subset of $X$. We develop the Walters' theory and discuss the existence of conformal measures in terms of the Perron-Frobenius-Ruelle operator for a continuous map…
A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using…
We present a rigorous analytic proof of a generalized continued fraction (GCF) identity for the transcendental constant $8/\pi^2$, a result recently conjectured via the algorithmic framework of the Ramanujan Machine. Distinct from canonical…
In this work we study qualitative properties of real analytic bounded maps. The main tool is approximation of real valued functions analytic in rectangular domains of the complex plane by continued g-fractions of Wall. As an application,…
We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations,…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and…
We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural…