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Aliasghar Sarizadeh

Publications and source records attributed to Aliasghar Sarizadeh.

6 recordsLinked to original sources
arXiv2026

Minimal Oscillation of Cesàro Averages Implies Non-Statisticality

We investigate the relationship between the global convergence of Cesàro averages and the pointwise statistical behavior of dynamical systems. First, we prove that if the Cesàro averages accumulate on at least two different measures (a property we call the minimal oscillation property) then the system is non-statistical. Second, we show that a system possesses a natural measure in the strong sense if and only if it is uniquely ergodic. As a consequence, every minimal homeomorphism on the circle possesses a natural measure in the strong sense which is physical and whose basin is the entire circle.

math.DS↗
arXiv2026

Equicontinuity of the Hutchinson operator $F$ and sensitivity of $F_-$

For an iterated function system $ \mathcal{F} = \{ f_1, \dots, f_k \} $ of homeomorphisms on a compact metric space $(X, d)$, write $ \mathcal{F}_-= \{ f_1^{-1}, \dots, f_k^{-1} \} $. The objective of this paper is to illustrate an iterated function system $\mathcal{F}$ of homeomorphisms on the circle that the Hutchinson operator of $\mathcal{F}$ is equicontinuous, but the Hutchinson operator of $\mathcal{F}_-$ is sensitive.

math.DS↗
arXiv2025

Attractors as a bridge from topological properties to long-term behavior in dynamical systems

This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions, including that the phase space of a topologically exact system is an attractor for its inverse, and vice versa, and that a system is topologically mixing if and only if its phase space is a physical attractor. Through iterated function systems (IFSs), we illustrate classes of non-trivial topologically mixing and topologically exact IFSs. Additionally, we use IFSs to provide an example of topologically mixing system, generated by finite of homeomorphisms on a compact metric space, that is not topologically exact. These findings connect topological properties with attractor types, providing deeper insights into the long-term dynamics of such systems.

math.DS↗
arXiv2023

Attractor for minimal iterated function systems

In the present work, we study the attractors of iterated function systems (IFSs) on connected and compact metric spaces. We prove that the whole of the phase space of a forward minimal IFS, for which some map admits an attracting fixed point, is an attractor.

math.DS↗
arXiv2015

Ergodicity of iterated function systems via minimality on the hyper spaces

We give a sufficient condition for the ergodicity of the Lebesgue measure for an iterated function system of diffeomorphisms. This is done via the induced iterated function system on the space of continuum (which is called hyper-space). We introduce a notion of minimality for induced IFSs which implies that the Lebesgue measure is ergodic for the original IFS. Here, to beginning, the required regularity is $C^1$. However, it is proven that the $C^1$-regularity is a redundant condition to prove ergodicity with respect to the class of quasi-invariant measures. As a consequence of mentioned results, we obtain ergodicity with respect to Lebesgue measure for several systems.

math.DS↗