The inverse Frobenius-Perron problem: A survey of solutions to the original problem formulation

<abstract><p>The inverse Frobenius-Perron problem (IFPP) is a collective term for a family of problems that requires the construction of an ergodic dynamical system model with prescribed statistical characteristics. Solutions to this problem draw upon concepts from ergodic theory and are scattered throughout the literature across domains such as physics, engineering, biology and economics. This paper presents a survey of the original formulation of the IFPP, wherein the invariant probability density function of the system state is prescribed. The paper also reviews different strategies for solving this problem and demonstrates several of the techniques using examples. The purpose of this survey is to provide a unified source of information on the original formulation of the IFPP and its solutions, thereby improving accessibility to the associated modeling techniques and promoting their practical application. The paper is concluded by discussing possible avenues for future work.</p></abstract>

ITERATED LOGARITHM SPEED OF RETURN TIMES

In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain $\unicode[STIX]{x1D709}$ depending on the dimension of the space, $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$ almost everywhere and $\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$ for almost all $x$ and $y$. This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.

Ergodic averages with prime divisor weights in

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$, $$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$ This answers a question raised by Cuny and Weber, who showed this result for $L^{p}$, $p>1$.

On the Large Deviations Theorem of Weaker Types

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e. type I, type I[Formula: see text], type II, type II[Formula: see text], type III, and type III[Formula: see text], and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large deviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I[Formula: see text] is ergodic, which is equivalent to the large deviations theorem of type II[Formula: see text] in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an [Formula: see text]-system.

Spectral properties of the Möbius function and a random Möbius model

Assuming Sarnak’s conjecture is true for any singular dynamical process, we prove that the spectral measure of the Möbius function is equivalent to Lebesgue measure. Conversely, under Elliott’s conjecture, we establish that the Möbius function is orthogonal to any uniquely ergodic dynamical system with singular spectrum. Furthermore, using Mirsky’s theorem, we find a new simple proof of Cellarosi–Sinai’s theorem on the orthogonality of the square of the Möbius function with respect to any weakly mixing dynamical system. Finally, we establish Sarnak’s conjecture for a particular random model.

Ergodic Axiom: The Ontological Mistakes in Economics

There are several ontological and consequently also methodological mistakes in contemporary mainstream economics. Among them, the so-called ergodic axiom is play significant role. It is understandable that the real economy elaborated as formalized mental model looks like dynamic system on first sight. However, that is right only of dynamical systems in mathematical formalism. Economy that is in our understanding societal and/or collective economy is complex evolving organism. If we imagine such organism in the form of dynamical system that is as clear mathematical formalism, we are losing their crucial authentic character. The significant irredeemable attribute of societal economy is lying in his complex evolving network process character created by large population of people with different decision-making and complex realizing among them. Going from these imaginations the two entities in a question that is dynamical system with their ergodicity and societal economic organism as complex evolving network are qualitative very different ones. That is the reason why we cannot accede with endeavours to draw on living economy straitjacket of ergodic axiom. To articulate that cause by other words ergodic dynamical systems are applicable for physical and partly for chemical entities and only scarcely are fit for living organisms. On the other hand however, as clear method the ergodic dynamical system have good applying for didactical approaches in economics where helping in better understanding some types of complexities in dynamics. The purpose of that essay is to discuss problems around usability of ergodic dynamical system theory and methods in economics in the age of advanced ICT knowledge based society.

On sums of indicator functions in dynamical systems

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every invertible ergodic dynamical system, for every increasing sequence (an)n∈ℕ⊂ℝ+ such that an↗∞ and an/n→0 as n→∞, there exists a dense Gδ of measurable sets A such that the sequence of the distributions of the partial sums $(\sfrac {1}{a_n})\sum _{i=0}^{n-1}(\ind _A-\mu (A))\circ T^i$ is dense in the set of the probability measures on ℝ.

Sparse Schrödinger Operators

We study spectral properties of a family [Formula: see text], indexed by a non-negative integer p, of one-dimensional discrete operators associated to an ergodic dynamical system (T,X,ℬ,μ) and defined for u in ℓ2(ℤ) and n in ℤ by [Formula: see text], where Vx(n)=f(Tnx) and f is a real-valued measurable bounded map on X. In some particular cases, we prove that the nature of the spectrum does not change with p. Applications include some classes of random and quasi-periodic substitutional potentials.

Almost everywhere convergence of convolution powers

Given an ergodic dynamical system (X,B,m, τ) and a probability measure μ on the integers, define for all f ∈ L1(X) The almost everywhere convergence of the convolution powers μnf(x) depends on the properties of μ. If μ has finite and then for all f ∈ Lp(X), 1< p < ∞, exists for a.e. x. However, if m2(μ) is finite and E(μ)≠0, then there exists E∈B such that a.e. and a.e. In the case when m2(μ) is infinite and E(μ)=0 we give examples for which we have divergence and other examples which show convergence is possible.