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Let $f : X\to X$ be a dominating meromorphic map on a compact K\"ahler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h^l_{\mathrm{top}}(f)$ for the action of $f$ on (local) analytic sets of dimension $0\leq l…

For a jointly measurable probability-preserving action $\tau:\mathbb{R}^D\curvearrowright (X,\mu)$ and a tuple of polynomial maps $p_i:\mathbb{R}\to \mathbb{R}^D$, $i=1,2,...,k$, the multiple ergodic averages \[ \frac{1}{T}\int_0^T…

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space…

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates $(x_n)$ given by $x_{n+1} = (I+\lambda_n A(\xi_{n+1},\,.\,))^{-1}(x_n)$ where $(A(s,\,.\,):s\in E)$ is a collection of maximal…

We present a novel formulation of ergodic trajectory optimization that can be specified over general domains using kernel maximum mean discrepancy. Ergodic trajectory optimization is an effective approach that generates coverage paths for…

We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic…

Proximal policy optimization (PPO) algorithm is a deep reinforcement learning algorithm with outstanding performance, especially in continuous control tasks. But the performance of this method is still affected by its exploration ability.…

In this manuscript, we consider finitely many maps, all of which are defined on a smooth compact measure space, with at least one map in the collection having degree strictly bigger than 1. Working with random dynamics generated by this…

Given an ergodic probability measure preserving dynamical system $\G\acts (X,\mu)$, where $\G$ is a finitely generated countable group, we show that the asymptotic growth of the number of finite models for the dynamics, in the sense of…

The maximum (or minimum) generalized eigenvalue of symmetric positive semidefinite matrices that depend on optimization variables often appears as objective or constraint functions in structural topology optimization when we consider…

Topological mapping of a large physical system on a graph, and its decomposition using universal measures is proposed. We find inherent limits to the potential for optimization of a given system and its approximate representations by…

In the variational approach to statistical mechanics, equilibrium states are the rigorous analogues of thermodynamic phases; the question of which invariant measures can arise as equilibrium states is therefore the question of which phases…

Let $\{H_{\lambda}\}$ be a continuous family of H\'{e}non maps parametrized by $\lambda\in M$, where $M\subset\mathbb C^k$ is compact. The purpose of this paper is to understand some aspects of the random dynamical system obtained by…

We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq…

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an…

It is commonly believed that Bayesian optimization (BO) algorithms are highly efficient for optimizing numerically costly functions. However, BO is not often compared to widely different alternatives, and is mostly tested on narrow sets of…

In this paper, we investigate capacity preserving transformations and their ergodicity. We show that for any measurable transformation $\theta$ there always exists a $\theta$-invariant capacity. We investigate some limit properties under…

Following Schweiger's generalization of multidimensional continued fraction algorithms, we consider a very large family of $p$-adic multidimensional continued fraction algorithms, which include Schneider's algorithm, Ruban's algorithms, and…

From a dynamical viewpoint, basic phase transitions of statistical mechanics can be regarded as a breaking of ergodicity. While many random models exhibiting such transitions at the thermodynamics limit exist, finite-dimensional examples…

We consider a nonstationary random walk on a compact metrizable abelian group. Under a classical strict aperiodicity assumption we establish a weak-* convergence to the Haar measure, Ergodic Theorem and Large Deviation Type Estimate.