arXiv subjects

Dawoud Ahmadi Dastjerdi

Publications and source records attributed to Dawoud Ahmadi Dastjerdi.

9 recordsLinked to original sources

arXiv2024

Attractor and its self-similarities for an IFS over arbitrary sub-shift

Consider a compact metric space $X$, and let $\mathcal{F}=\{f_1,\,f_2,\ldots,\, f_k\}$ be a set of contracting and continuous self maps on $X$. Let $Σ$ be a sub-shift on $k$ symbols, and let $Σ_k$ be the full shift. Define $\mathcal{L}_n(Σ)$ as the set of words of length $n$ in $Σ$. For $u=u_1\cdots u_n\in \mathcal{L}_n(Σ)$, set $f_u:=f_{u_n}\circ\cdots \circ f_{u_1}$ and $H^n(\cdot):=\cup_{u \in \mathcal{L}_{n}(Σ_k)} f_{u}(\cdot)$. When $Σ=Σ_k$, $H^n(\cdot)$ is the $n$th iteration of the Hutchinson's operator, and there exists a compact set $S= \lim_{n \rightarrow \infty} H^n(A)$ for any compact $A\subseteq X$ with $H^n(S)=S$ (self-similarity criteria) for $n\in\N$. For arbitrary $Σ$, the above limit exists; but it is not necessarily true that $H^n(S)=S$. Sufficient conditions on $Σ$ are provided to have $H^n(S)=S$ for all or some $n\in\N$, and then the dynamics of $S$ under the admissible iterations of $f_i$'s defined by $Σ$ are investigated.

math.DS↗

arXiv2023

Topological conjugacy and its relations for symbolic matrices

In 1988 Boyle and Krieger defined sub-matrices for representation matrices of sofic shift. This paper presents some details of relations between integral sub-matrices and representation matrices. Besides, we express a new version of the Decomposition Theorem by sub-matrices. Generally, strong shift equivalence (conjugacy) of sub-matrices does not apply to representation matrices, but we show that this result can be achieved by the fixed diagonal integral sub-matrix.

math.DS↗

arXiv2022

Shift limits of a non-autonomous system

Let $t=t_1t_2\cdots$ be an element of the full shift with shift map $τ$ on a finite set of characters $\mathcal{A}$ and let $ Σ=\text{ closure} \{τ^i(t):\;i\in\N\cup\{0\}\}$. Let $f_t=f_{t_1,\,\infty}=\cdots\circ f_{t_2}\circ f_{t_1} $ be a non-autonomous system over a compact metric space $ X $ where $t_i\in \mathcal A $. The set $\F_t^+=\{f_{τ^i(t)}:\; i\in\N\}$ is called the shifted family of $f_t$. If $t$ is a transitive point of the full shift on $\mathcal A$, then by introducing a natural topology, $\overline{\F_t^+}$ is a classical IFS; otherwise, $\overline{\F_t^+}=\{f_σ=f_{σ_1,\,\infty}:\; σ\inΣ\}$ is a generalized IFS. We will show that if $ f_t$ has some various shadowing and specification properties, then this is true for $f_σ\in\overline{\F^+_t}$; however, this claim is not true for other properties such as transitivity, mixing and exactness. Also, if $ Σ$ is sofic and $x\in X$ is periodic point for some $f_σ\in\overline{\F^+_t}$, then there is a periodic $σ'\inΣ$ such that $x$ is periodic for $f_{σ'}\in\overline{\F^+_t}$.

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arXiv2022

Iterated function systems over arbitrary shift spaces

The orbit of a point $x\in X$ in a classical iterated function system (IFS) can be defined as $\{f_u(x)=f_{u_n}\circ\cdots \circ f_{u_1}(x):$ $u=u_1\cdots u_n$ is a word of a full shift $Σ$ on finite symbols and $f_{u_i}$ is a continuous self map on $X$ $\}$. One also can associate to $σ=σ_1σ_2\cdots\inΣ$ a non-autonomous system $(X,\,f_σ)$ where the trajectory of $x\in X$ is defined as $x,\,f_{σ_1}(x),\,f_{σ_1σ_2}(x),\ldots$.Here instead of the full shift, we consider an arbitrary shift space $Σ$. Then we investigate basic properties related to this IFS and the associated non-autonomous systems. In particular, we look for sufficient conditions that guarantees that in a transitive IFS one may have a transitive $(X,\,f_σ)$ for some $σ\inΣ$ and how abundance are such $σ$'s.

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arXiv2017

Types of mixings and transitivities in topological dynamics

Using the combinatorial properties of subsets of integers, a classification of metric dynamical systems was given in [V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure-preserving systems, Colloquium Mathematicum, 110(1), (2008), 117-150]. As a result, some new families emerged. Here, their counterparts in topological dynamics has been considered. The differences will be discussed and new classes of systems in topological dynamics will be introduced.

math.DS↗

arXiv2015

Mixing coded systems

We show that a coded system is mixing if and only if it is totally transitive and if in addition it has a relatively prime generator, then it has strong property P. We continue by showing that a mixing half-synchronized system has such a generator. Moreover, we give an example of a mixing coded system which does not have any relatively prime generator.

math.DS↗

arXiv2015

On semi-open codes and bi-continuing almost everywhere codes

We will show that a system is synchronized if and only if it has a cover whose cover map is semi-open. Also, any factor code on an irreducible sofic shift is semi-open and the image of a synchronized system by a semi-open code is synchronized. On the other side, right-closing semi-open extension of an irreducible shift of finite type is of finite type. Moreover, we give conditions on finite-to-one factor codes to be open and show that any semi-open code on a synchronized system is bi-continuing a.e.. We give some sufficient conditions for a right-continuing a.e. factor code being right-continuing everywhere.

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arXiv2011

Characterization of Entropy for Spacing shifts

Suppose $P\subseteq \mathbb{N}$ and let $(Σ_P,\,σ_P)$ be the space of a spacing shift. We show that if entropy $h_{σ_P}=0$ then $(Σ_P,\,σ_P)$ is proximal. Also $h_{σ_P}=0$ if and only if $P=\mathbb N\setminus E$ where $E$ is an intersective set. Moreover, we show that $h_{σ_P}>0$ implies that $P$ is a $Δ^*$ set; and by giving a class of examples, we show that this is not a sufficient condition. Then there is enough results to solve question 5 given in [J. Banks et al., \textit{Dynamics of Spacing Shifts}, Discrete Contin. Dyn. Syst., to appear.].

math.DS↗

arXiv2011

Generating Functions for Special Flows over the 1-Step Countable Topological Markov Chains

Let $Y$ be a topological Markov chain with finite leading and follower sets. Special flow over $Y$ whose height function depends on the time zero of elements of $Y$ is constructed. Then a formula for computing the entropy of this flow will be given. As an application, we give a lower estimate for the entropy of a class of geodesic flows on the modular surface. We also give sufficient conditions to guarantee the existence of a measure with maximal entropy.

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