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In this article, we develop a functional-analytic framework to establish existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states for a broad class of dynamical systems, potentially discontinuous…
We survey an area of recent development, relating dynamics to theoretical computer science. We discuss the theoretical limits of simulation and computation of interesting quantities in dynamical systems. We will focus on central objects of…
We present analytical treatment of quantum walks on multidimensional hyper-cycle graphs. We derive the analytical expression of the probability distribution for strong and weak decoherence regimes. Upper bound to mixing time is obtained.
Generic quantum systems --as much as their classical counterparts-- pass arbitrarily close to their initial state after sufficiently long time. Here we provide an essentially exact computation of such recurrence times for generic…
A new method of deriving reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a set of resolved variables that define a model reduction, the quasi-equilibrium…
Analyzing the dynamics of open quantum systems has a long history in mathematics and physics. Depending on the system at hand, basic physical phenomena that one would like to explain are, for example, convergence to equilibrium, the…
In this paper we study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy…
The main purpose of this article is to study box dimension of orbits near hyperbolic and nonhyperbolic fixed points of discrete dynamical systems in higher dimensions. We generalize the known results for one-dimensional systems, that is,…
We present an inverse method to construct large classes of chaotic invariant sets together with their exact statistics. The associated dynamical systems are characterized by a probability distribution and a two-form. While our emphasis is…
The question of how irreversibility can emerge as a generic phenomena when the underlying mechanical theory is reversible has been a long-standing fundamental problem for both classical and quantum mechanics. We describe a mechanism for the…
In the classical theorems of extreme value theory the limits of suitably rescaled maxima of sequences of independent, identically distributed random variables are studied. So far, only affine rescalings have been considered. We show,…
We analyze the stability of general nonlinear discrete-time stochastic systems controlled by optimal inputs that minimize an infinite-horizon discounted cost. Under a novel stochastic formulation of cost-controllability and detectability…
Extreme value analysis is an essential methodology in the study of rare and extreme events, which hold significant interest in various fields, particularly in the context of environmental sciences. Models that employ the exceedances of…
An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak…
A general analytical approach to the statistical description of quantum graph spectra based on the exact periodic orbit expansions of quantum levels is discussed. The exact and approximate expressions obtained in \cite{Anima} for the…
We study distribution of orbits sampled at polynomial times for uniquely ergodic topological dynamical systems $(X, T)$. First, we prove that if there exists an increasing sequence $(q_n)$ for which the rigidity condition \[…
The stability theory for hyperbolic initial boundary value problems relies most of the time on the Laplace transform with respect to the time variable. For technical reasons, this usually restricts the validity of stability estimates to the…
The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic dynamics developed by the first author \textit{et al}. to establish quenched versions of the large deviation…
A one-dimensional quantum oscillator is monitored by taking repeated position measurements. As a first con- tribution, it is shown that, under a quantum nondemolition measurement scheme applied to a system initially at the ground state, (i)…
The block maximum method, which is widely used in extreme value analysis, uses a generalized extreme value distribution to approximate that of the maximum of m observations. The quality of this approximation depends on the value of m and…