arXiv subjects

Nima Alibabaei

Publications and source records attributed to Nima Alibabaei.

7 recordsLinked to original sources

arXiv2026

Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets

We establish a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets. The resulting formula extends the Kenyon--Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.

math.DS↗

arXiv2026

Lyapunov exponents for uniformly hyperbolic random matrix products

We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the matrices are projectively uniformly hyperbolic with respect to the Markov shift, then $\lambda$ admits an explicit representation in terms of an infinite matrix. This rapidly convergent representation yields a polynomial-time algorithm for approximating $\lambda$: only $O\big( (\log(1/\varepsilon))^3 \big)$ arithmetic operations are needed to achieve error $\varepsilon$. Furthermore, $\lambda$ depends real analytically on the matrix entries and the transition probabilities near a projectively uniformly hyperbolic system, and each Taylor coefficient can be approximated in polynomial time.

math.DS↗

arXiv2026

Lyapunov exponents for random products of non-negative matrices

We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite matrix. We further show that non-negative invertible $2 \times 2$ matrices are simultaneously conjugate to positive matrices if and only if ``generalized'' heteroclinic connections do not occur among products of length at most $2$. These results yield a series formula for the Hausdorff dimension of the intersection of the middle-$n$th Cantor set with a random translate of itself, for every natural number $n$ except $4$. Furthermore, our method applies to the intersection of thick Cantor sets under random translation. We also determine the almost sure growth rate of i.i.d. three-term recurrences with finitely many positive coefficients.

math.DS↗

arXiv2025

On the intersection of Cantor sets and products of random matrices

Kenyon and Peres (1991) showed that the Hausdorff dimension of intersections of randomly translated Cantor sets can be expressed in terms of the top Lyapunov exponent of a product of random matrices, and this exponent can be written as an integral with respect to stationary measures on the projective line. Although explicit computations are available when stationary measures are discrete, the continuous case has remained challenging. In this paper we introduce new combinatorial and analytic tools that allow us to compute the Lyapunov exponent, and hence the Hausdorff dimension, in a broad class of examples where stationary measures are continuous. As an application, we complete the dimension computation in the setting where a single digit is forbidden; for example, we determine the Hausdorff dimension of the intersection of the middle-seventh Cantor set with a random translate of itself.

math.DS↗

arXiv2024

Exact Hausdorff dimension of some sofic self-affine fractals

Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial cases, we demonstrate that the dimension can in fact be calculated explicitly. Specifically, the dimension is expressed as the solution to an infinite-degree equation with explicit coefficients, which also corresponds to the spectral radius of a certain linear operator. Our result provides the first non-trivial calculation of the exact Hausdorff dimension of sofic sets in $\mathbb{R}^3$. This is achieved by developing a new technique inspired by the work of Kenyon and Peres (1998).

math.DS↗

arXiv2024

Improved dimension theory of sofic self-affine fractals

Follow-up comment by the author: Theorem 2.2 in this paper is a special case of Theorems 1.1 and 4.1 in the article "Weighted thermodynamic formalism on subshifts and applications", Asian J. Math. 16 (2012), by J. Barral and D. J. Feng. In addition, Zhou Feng studied the conditions under which general self-affine fractals, including sofic sets, have the same Hausdorff dimension and box dimension in the paper "On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets", arXiv:2405.03213. I would like to thank Dr. Zhou Feng for pointing out these works. The calculation of the exact Hausdorff dimension of sofic sets presented in this article is refined in my subsequent work "Exact Hausdorff dimension of some sofic self-affine fractals", arXiv:2412.05805. Original abstract: We establish a combinatorial expression for the Hausdorff dimension of a given self-affine fractal in any Euclidean space. This formula includes the extension of the work by Kenyon and Peres (1996) on planar sofic sets and yields an exact value for the dimension of certain sofic sets in $\mathbb{R}^3$ or higher. We also calculate the Minkowski dimension of sofic sets and establish a sufficient and presumably necessary condition for planar sofic sets to have the same Minkowski and Hausdorff dimension. The condition can be regarded as a generalization of the classical result for Bedford-McMullen carpets.

math.DS↗

arXiv2023

Weighted topological pressure revisited

Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto (2022) redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto's approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

math.DS↗