arXiv subjects

Haritha Cheriyath

Publications and source records attributed to Haritha Cheriyath.

6 recordsLinked to original sources

arXiv2025

A note on the perturbations of subshifts

In this paper, we consider different classes of subshifts and study their perturbations obtained by forbidding sequences that contain a given word as a subword. We show that the perturbations of sofic shifts are sofic. Though not true for general coded systems, we provide sufficient conditions under which perturbations of synchronized systems remain synchronized. We investigate which systems remain conjugate under perturbation by a word from a coded system. We also explicitly calculate the drop in entropy under perturbation when the subshift is an $\mathcal{S}$-gap shift, and obtain its exponential decay as the length of the word increases. In the case of subshifts of finite type, we explore the drop in entropy under multi-word perturbations.

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arXiv2025

Asymptoticity, automorphism groups and strong orbit equivalence

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal{S}$-adic subshifts. We derived the uncountable case by showing that any topological dynamical system with countably many asymptotic components has zero topological entropy. We also construct systems with arbitrarily high subexponential word complexity with only one asymptotic class. We deduce that within any strong orbit equivalence class, there exists a subshift whose automorphism group is isomorphic to $\mathbb{Z}$.

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arXiv2024

On Escape rate for subshift with Markov measure

In this paper, we consider a subshift of finite type with Markov measure. By considering a union of cylinders as holes, we investigate the exponential growth rate of measure of points whose orbits do not escape into the hole over a fixed number of iterations. We present two formulations for this escape rate: one based on the spectral radius of the Hadamard product of a related adjacency matrix and the stochastic matrix with respect to which the Markov measure is defined, and the other utilizing a recurrence relation. These formulations enable a comparative analysis of escape rates into distinct holes.

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arXiv2023

A combinatorial approach to study subshifts associated with multigraphs

A subshift of finite type over finitely many symbols can be described as a collection of all infinite walks on a digraph with at most a single edge from a vertex to another. The associated finite set $\F$ of forbidden words is a constraint which determines the language of the shift entirely. In this paper, in order to describe infinite walks on a multigraph, we introduce the notion of multiplicity of a word (finite walk) and define repeated words as those having multiplicity at least $2$. In general, for given collections $\F$ of forbidden words and $\R$ of repeated words with pre-assigned multiplicities, we define notion of a generalized language which is a multiset. We obtain a subshift associated with $\F$ and $\R$ such that its entropy is calculated using the generalized language. We also study the relationship between the language of this subshift and the generalized language. We then obtain a combinatorial expression for the generating function that enumerates the number of words of fixed length in this generalized language. This gives the Perron root and eigenvectors of the adjacency matrix with integer entries associated to the underlying multigraph. Using this, the topological entropy and an alternate definition of Parry measure for the associated edge shift are obtained. We also discuss some properties of Markov measures on this subshift.

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arXiv2022

On the Perron root and eigenvectors associated with a subshift of finite type

In this paper, we describe the relationship between the Perron root and eigenvectors of an irreducible subshift of finite type with the correlation between the forbidden words in the subshift. In particular, we derive an expression for the Perron eigenvectors of the associated adjacency matrix. As an application, we obtain the Perron eigenvectors for irreducible $(0,1)$ matrices which are adjacency matrices for directed graphs. Moreover, we derive an alternate definition of the Parry measure in ergodic theory on an irreducible subshift of finite type.

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arXiv2022

Subshifts of Finite Type with a Hole

This paper examines the relationship between the escape rate and the minimal period of the hole. We consider a subshift of finite type on $q$ symbols with a union of $t$ cylinders based at words of identical length $p$ as the hole. The escape rate into the hole relates to the asymptotic behavior of the number of words of fixed length that do not contain the fixed set of (forbidden) words at which the cylinders are based. We explore the relationship between the escape rate and $r(z)$, a rational function of the correlations between the forbidden words. In particular, we prove that there exists a constant $D(t,p)$ such that if $q>D(t,p)$, then the escape rate is faster for the hole with larger value of $r(z)$ evaluated at $D(t,p)$. Further, we consider two holes each of which is a union of cylinders based at words of equal length, having zero cross-correlation, and prove that the larger is the minimal period of the collection, the faster is the escape rate. However, when the cross-correlations are non-zero, we give examples to prove that this result fails to hold. Our results are more general than the existing ones known for maps conjugate to a full shift with a single cylinder as the hole. The existing results arise as a special case of our results.

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