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Abbas Fakhari

Publications and source records attributed to Abbas Fakhari.

9 recordsLinked to original sources
arXiv2026

Intermingled basins: Kan's example on the Riemann sphere

Kan's discovery of dynamical systems with two attractors whose basins of attraction both have full support, featured specific examples of skew product systems of interval diffeomorphisms forced by expanding circle maps. The interval diffeomorphisms are polynomial maps and can hence be considered on the Riemann sphere. We consider the resulting skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, and establish the existence of three attractors whose basins of attraction all have full support and are thus intermingled.

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arXiv2026

Generalized Horseshoe Maps part I: limit laws and absolute continuity of physical measure

We apply thermodynamic formalism to a generalized horseshoe map. We prove that a tailored anisotropic Banach space with weighted norms yields a spectral gap for the transfer operator, implying the existence of a unique physical measure and limit theorems. Under the virtually expanding condition, this measure is absolutely continuous with respect to Lebesgue measure, with density in the Sobolev space $H^\mu$, for some $\mu<1/2$.

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arXiv2024

Stable local dynamics: expansion, quasi-conformality and ergodicity

In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class $C^{1+α}$. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.

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arXiv2024

Thick attractors with intermingled basins

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop theory for it and use it in particular to provide examples of thick metric attractors with intermingled basins.

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arXiv2018

Ergodicity of non-autonomous discrete systems with non-uniform expansion

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $C^{1+α}$ local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal $C^{1+α}$ local diffeomorphisms of a compact manifold is Lebesgue ergodic.

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arXiv2018

Expanding actions: minimality and ergodicity

We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+α}$ conformal) is robustly minimal (resp. ergodic with respect to Lebesgue measure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.

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arXiv2017

Connectedness of the set of central Lyapunov exponents

We show that there is a residual subset $\mathcal{R}$ of $Diff^1(M)$ such that for any $f\in\mathcal{R}$ and any partially hyperbolic homoclinic class $H(p,f)$ with one dimensional center direction, the set of central Lyapunov exponents associated with the ergodic with either full support or positive entropy is an interval.

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arXiv2015

Density of fiberwise orbits in minimal iterated function systems on the circle

We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.

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