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Meysam Nassiri

Publications and source records attributed to Meysam Nassiri.

9 recordsLinked to original sources
arXiv2026

Cocycles with Quasi-Conformality I: Stability and abundance

This is the first part of a series of papers devoted to the study of linear cocycles over chaotic systems. In the present paper, we establish the existence of such cocycles that $\mathcal{C}^α$-stably exhibit fiberwise bounded orbits ($α>0$). The proof is based on a new mechanism which yields stable elliptic-type behavior in $\mathrm{GL}(d,\mathbb{R})$ or $\mathrm{SL}(d,\mathbb{R})$ cocycles. Moreover, we show that this phenomenon is $\mathcal{C}^0$-dense among $\mathrm{SL}(d,\mathbb{R})$ cocycles over a shift of finite type without dominated splitting.

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arXiv2026

Cocycles with Quasi-Conformality II: Ergodic measures with positive entropy

As the second part of a series on linear cocycles over chaotic systems, this paper establishes a "multiple covering principle" that robustly yields positive-entropy ergodic measures supported on fiberwise uniformly bounded orbits. Using this mechanism, we prove that any continuous $\mathrm{SL}(d,\mathbb{R})$ cocycle over a positive-entropy subshift of finite type either admits a dominated splitting or can be $\mathcal{C}^0$-approximated by one that \emph{stably} supports such measures. The stability holds in Hölder topology which is optimal. Additionally, for non-isometric cocycles, we show that the topological entropy of these bounded orbits is strictly less than that of the base subshift.

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arXiv2026

Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections

It is well known that a pair of compact sets in $\mathbb{R}^d$ ($d \in \mathbb{N}$) can be separated by small deformations if the sum of their upper box dimensions is less than $d$. In this paper, we demonstrate that this dimension constraint is optimal for regular Cantor sets. Specifically, for any prescribed upper box dimensions whose sum is greater than $d$, we construct classes of pairs of regular Cantor sets that exhibit $C^{1+α}$-stable intersections. Our method is geometrically flexible, enabling the construction of examples with arbitrarily small thickness in both projectively hyperbolic and nearly conformal regimes. These results also extend to the complex setting for holomorphic Cantor sets in $\mathbb{C}^d$. The proof relies on the "covering criterion" for stable intersection introduced in the first part of this series [NZ25], which generalizes the "recurrent compact set criterion" of Moreira-Yoccoz to higher dimensions.

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arXiv2026

Cantor sets in higher dimension I: Criterion for stable intersections

We study the geometry of dynamically defined Cantor sets in arbitrary dimensions, introducing a criterion for $\mathcal{C}^{1+α}$ stable intersections of such Cantor sets, under a mild bunching condition. This condition is naturally satisfied for perturbations of conformal Cantor sets and, in particular, always holds in dimension one. Our work extends the celebrated recurrent compact set criterion of Moreira and Yoccoz for stable intersection of Cantor sets in the real line to higher-dimensional spaces. Based on this criterion, we develop a method for constructing explicit examples of stably intersecting Cantor sets in any dimension. This construction operates in the most fragile and critical regimes, where the Hausdorff dimension of one of the Cantor sets is arbitrarily small and both Cantor sets are nearly homothetical. All results and examples are provided in both real and complex settings.

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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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arXiv2012

Prime ends rotation numbers and periodic points

We study the problem of existence of a periodic point in the boundary of an invariant domain for a surface homeomorphism. In the area-preserving setting, a complete classification is given in terms of rationality of Carathéordory's prime ends rotation number, similar to Poincaré's theory for circle homeomorphisms. In particular, we prove the converse of a classic result of Cartwright and Littlewood. This has a number of consequences for generic area preserving surface diffeomorphisms. For instance, we extend previous results of J. Mather on the boundary of invariant open sets for $C^r$-generic area preserving diffeomorphisms. Most results are proved in a general context, for homeomorphisms of arbitrary surfaces with a weak nonwandering-type hypothesis. This allows us to prove a conjecture of R. Walker about co-basin boundaries, and it also has applications in holomorphic dynamics.

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arXiv2011

Robust Transitivity in Hamiltonian Dynamics

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting "large" robustly transitive sets. We show that the $C^\infty$ closure of such open sets contains a variety of systems, including so-called a priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the proof is a combination of studying minimal dynamics of symplectic iterated function systems and a new tool in Hamiltonian dynamics which we call symplectic blender.

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