arXiv subjects

Reza Mohammadpour

Publications and source records attributed to Reza Mohammadpour.

13 recordsLinked to original sources

arXiv2025

Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications

We contribute to the thermodynamic formalism of Hölder continuous fiber-bunched matrix cocycles, Anosov diffeomorphisms, and hyperbolic repellers. Specifically, we prove that $1$-typical fiber-bunched cocycles $\mathcal{A}$ over topologically mixing subshifts of finite type admit a unique Gibbs equilibrium state $μ_t$ associated with the non-additive family of potentials $\{t \log \|\mathcal{A}^n\|\}_{n \in \mathbb{N}}$, for a range of parameters $t \in (-t_*, +\infty)$, where $t_* > 0$. Furthermore, these equilibrium states are $ψ$-mixing, therefore weak Bernoulli. In addition, these results allow us to derive consequences for the thermodynamic formalism of open sets of hyperbolic repellers and Anosov diffeomorphisms. In particular, it provides a positive answer to a conjecture posed by Gatzouras and Peres for $C^1$-open sets of $α$-fiber-bunched hyperbolic repellers.

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arXiv2025

Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$

We consider a one-step matrix cocycle generated by a pair of non-negative parabolic matrices and study the equilibrium measures for $t\log \|\mathcal A\|$ as $t$ runs over the reals. We show that there is a freezing first order phase transition at some parameter value $t_c$ so that for $t t_c$, the equilibrium measure is unique, non-atomic and fully supported. The phase transition closely resembles the classical Hofbauer example. In particular, our example shows that there may be non-unique equilibrium measures for negative $t$ even if the cocycle is strongly irreducible and proximal.

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arXiv2025

Periodic approximation of topological Lyapunov exponents and the joint spectral radius for cocycles of mapping classes of surfaces

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.

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arXiv2025

Multifractal formalism of Lyapunov exponents for fiber-bunched linear cocycles

We develop a higher-dimensional extension of multifractal analysis for typical fiber-bunched linear cocycles. Our main result is a relative variational principle, which shows that the topological entropy of Lyapunov exponent level sets can be approximated by the metric entropy of ergodic measures fully concentrated on those level sets, addressing a question posed by Breuillard and Sert. We also establish a variational principle for the generalized singular value function. As an application to dynamically defined linear cocycles, we obtain a multifractal formalism for open sets of $C^{1+\alpha}$ repellers and Anosov diffeomorphisms.

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arXiv2025

Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles

In this paper we study ergodic optimization problems for typical cocycles. We consider one-step $SL(2,\mathbb{R})$-cocycles that satisfy pinching and twisting conditions. We prove that the Lyapunov maximizing measures have zero entropy under additional assumptions that the maps $e_1$ and $e_2$ are one-to-one on the Mather set.

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arXiv2024

Birkhoff spectrum for diagonally self-affine sets and digit frequencies for GLS systems with redundancy

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine IFS's. Also, we study Besicovitch-Eggleston sets for finite GLS number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

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arXiv2024

Entropy spectrum of Lyapunov exponents for typical cocycles

In this paper, we study the size of the level sets of all Lyapunov exponents. For typical cocycles, we establish a variational relation between the topological entropy of the level sets of Lyapunov exponents and the topological pressure of the generalized singular value function.

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arXiv2024

Uniform quasi-multiplicativity of locally constant cocycles and applications

In this paper, we show that a locally constant cocycle $\mathcal{A}$ is $k$-quasi multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that $\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $ψ$-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets.

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arXiv2023

Restricted variational principle of Lyapunov exponents for typical cocycles

In this paper, we study the multifractal formalism of Lyapunov exponents for typical cocycles. We establish a variational relation between the Legendre transform of topological pressure of the generalized singular value function and measure-theoretic entropies. As a consequence, we show that the restricted variational principle of Lyapunov exponents holds for typical cocycles.

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arXiv2022

Lyapunov spectrum properties and continuity of the lower joint spectral radius

We study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show the continuity of the entropy spectrum at the boundary of Lyapunov spectrum in the sense that $h_{top}(E(α_{t}))\ \rightarrow h_{top}(E(β(\mathcal{A}))$ for generic cocycles, where $E(α)=\{x\in X: \lim_{n\rightarrow \infty}\frac{1}{n}\log \|\mathcal{A}^{n}(x)\|=α\}$. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles over mixing subshifts of finite type. Moreover, we prove the restricted variational principle for such cocycles. We prove the continuity of the lower joint spectral radius for general cocycles under the assumption that linear cocycles satisfy a cone condition.

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arXiv2021

On positive Lyapunov exponents and SRB measures for partially hyperbolic systems

In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds admitting a dominated splitting $E^{cs} \oplus E^{cu}$. First, we prove that the smallest Lyapunov exponent along $E^{cu}$, computed with respect to the Lebesgue measure, is computable using observable measures. Then we show that if the Lyapunov exponents along $E^{cu}$ are positive Lebesgue almost everywhere and $E^{cu}$ admits a finest 1-dominated splitting on the support of an ergodic observable measure then $f$ is non-uniformly expanding along $E^{cu}$. As a byproduct, every $C^{1+\alpha}$ diffeomorphism exhibiting a dominated splitting $E^{s} \oplus E^{cu}$ where $E^{cu}$ fulfills the previous assumptions admits an SRB measure.

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arXiv2021

Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $Λ$ over the circle embedded in $\mathbb{R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing \emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$ strong stable foliation, we prove that Hausdorff dimension ${\rm HD}(Λ\cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else ${\rm HD}(Λ)=t_0+1$. We prove also that for the packing measure $0<Π_{t_0}(Λ\cap W^s)<\infty$ but for Hausdorff measure ${\rm HM}_{t_0}(Λ\cap W^s)=0$ for all $W^s$. Also $0<Π_{1+t_0}(Λ) <\infty$ and ${\rm HM}_{1+t_0}(Λ)=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure ${\rm HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.

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arXiv2020

Zero temperature limits of equilibrium states for subadditive potentials and approximation of the maximal Lyapunov exponent

In this paper we study ergodic optimization problems for subadditive sequences of functions on a topological dynamical system. We prove that for $t\rightarrow \infty$ any accumulation point of a family of equilibrium states is a maximizing measure. We show that the Lyapunov exponent and entropy of equilibrium states converge in the limit $t\rightarrow \infty$ to the maximum Lyapunov exponent and entropy of maximizing measures. In the particular case of matrix cocycles we prove that the maximal Lyapunov exponent can be approximated by Lyapunov exponents of periodic trajectories under certain assumptions.

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