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Let f:[0, 1]→[0, 1] be a multimodal map with positive topological entropy. The dynamics of the renormalization operator for multimodal maps have been investigated by Daniel Smania. It is proved that the measure of maximal entropy for a specific category of Cr interval maps is unique.
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (eg, an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.
Abstract It is known that if each pre-orbit of a non-injective endomorphism is dense, the endomorphism is transitive (i.e., a dense orbit exists). However, it is still unknown whether the pre-orbits of an Anosov map are dense, and the conditions necessary for all pre-orbits to be dense are also unknown. Using the properties of integral lattices, we construct our proof by considering the pre-orbits of linear endomorphisms. We introduce a class of hyperbolic linear endomorphisms characterized by the property of absolute hyperbolicity to prove that if A : Tm → Tm is an absolutely hyperbolic endomorphism, the pre-orbit of any point is dense in Tm.
The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the averag
In this paper we give a classification of special endomorphisms of nil-manifolds: Let be a covering map of a nil-manifold and denote by the nil-endomorphism which is homotopic to . If is a special -map, then is a hyperbolic nil-endomorphism and is topologically conjugate to .
In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph $\Gamma $, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of $\Gamma $ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular-digraph $\Gamma $ with diameter $ D= 2$, degree $ k\leq 3$ or $\lambda= 0$($\lambda $ is the number of 2-paths from $ u $ to $ v $ for an edge $ uv $ of $\Gamma $) is super vertex-connected, that is, any minimum vertex cut of $\Gamma $ is the set of all out-neighbors (or in-neighbors) of a single vertex in $\Gamma $. These results ext
We introduce shadowing (specification property, stroboscopical property) for dynamical systems on uniform space. Our focuses on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space with at least two elements, a nonempty set and a self--map the generalized shift dynamical system : has shadowing property, has (almost) weak specification property if and only if does not have any periodic point, has (uniform) stroboscopical property if and only if is one-to-one, has strongly stroboscopical property (resp. specification property) if and only if is one-to-one without any periodic point.Subjects: D
In this paper, we show that the edge connectivity of a distance-regular digraph with valency is and for , any minimum edge cut of is the set of all edges going into (or coming out of) a single vertex. Moreover we show that the same result holds for strongly regular digraphs. These results extend the same known results for undirected case with quite different proofs.
For an endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. But the converse has not been investigated before. Here we are going to show that it is true for Anosov Endomorphisms on closed manifolds, by the fact that Anosov endomorphisms are covering maps.
In this paper we give a classification of special endomorphisms of nil-manifolds: Let be a covering map of a nil-manifold and denote by the nil-endomorphism which is homotopic to . If is a special -map, then is a hyperbolic nil-endomorphism and is topologically conjugate to .
For an endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. The inverse has been shown for a residual set of points but the the exact inverse has not yet been investigated before. Here we are going to show that under some conditions it is true for Anosov endomorphisms on closed manifolds, by using the fact that Anosov endomorphisms are covering maps.Subjects: Dynamical Systems (math. DS)Cite as: arXiv: 1702.08167 [math. DS](or arXiv: 1702.08167 v4 [math. DS] for this version)Submission historyFrom: Khosro Tajbakhsh [view email][v1] Mon, 27 Feb 2017 07: 34: 19 GMT (9kb)[v2] Tue, 28 Feb 2017 14: 40: 00 GMT (93kb, D)[v3] Mon, 27 Mar 2017 18: 22: 39 GMT (94kb, D)[
In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space with at least two elements, a nonempty set and a self--map the generalized shift dynamical system :\begin {itemize}\item has (almost) weak specification property if and only if does not have any periodic point,\item has (uniform) stroboscopical property if and only if is one-to-one.\end {itemize}
In the following text we introduce specification property (stroboscopical property) for dynamical systems on uniform space. We focus on two classes of dynamical systems: generalized shifts and dynamical systems with Alexandroff compactification of a discrete space as phase space. We prove that for a discrete finite topological space with at least two elements, a nonempty set and a self--map the generalized shift dynamical system :\begin {itemize}\item has (almost) weak specification property if and only if does not have any periodic point,\item has (uniform) stroboscopical property if and only if is one-to-one.\end {itemize}
In this paper we give a classification of special endomorphisms of infra-nil-manifolds: Let f: N/Γ→ N/Γ be a covering map of an infra-nil-manifold and denote by A: N/Γ→ N/Γ the infra-nil-endomorphism which is homotopic to f. If f is a special TA-map, then A is a hyperbolic infra-nil-endomorphism and f is topologically conjugate to A.
For an endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. But the converse has not been investigated before. Here we are going to show that it is true for Anosov Endomorphisms on closed manifolds, by the fact that Anosov endomorphisms are covering maps.
In this paper we give a classification of special endomorphisms of nil-manifolds: Let be a covering map of a nil-manifold and denote by the nil-endomorphism which is homotopic to . If is a special -map, then is a hyperbolic nil-endomorphism and is topologically conjugate to .
In this paper the notion of asymptotic measure expansiveness is introduced and its relationship with dominated splitting is considered. It is proved that if a diffeomorphism admits a co-dimension one dominated splitting then it is asymptotic measure expansive. Also, a diffeomorphism with a homoclinic tangency can be perturbed to a non-asymptotic measure expansive diffeomorphism.