scholarly journals No semiconjugacy to a map of constant slope

2014 ◽  
Vol 36 (3) ◽  
pp. 875-889 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Sufficient Conditions ◽  
Sufficient Criterion ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Piecewise Continuous ◽  
Necessary And Sufficient ◽  
Constant Slope

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

scholarly journals Constant slope maps on the extended real line

2017 ◽  
Vol 38 (8) ◽  
pp. 3145-3169 ◽  
Author(s):  
MICHAŁ MISIUREWICZ ◽  
SAMUEL ROTH
Keyword(s):  
Real Line ◽  
Half Line ◽  
Piecewise Monotone ◽  
Bounded Interval ◽  
Interval Map ◽  
Piecewise Continuous ◽  
Constant Slope

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

scholarly journals Necessary and sufficient criterion for k-separability of N-qubit noisy GHZ states

2018 ◽  
Vol 16 (04) ◽  
pp. 1850037 ◽  
Author(s):  
Xiao-Yu Chen ◽  
Li-Zhen Jiang ◽  
Zhu-An Xu
Keyword(s):  
White Noise ◽  
Sufficient Conditions ◽  
Ghz State ◽  
Entangled State ◽  
Multipartite Entanglement ◽  
Quantum Code ◽  
Sufficient Criterion ◽  
Output State ◽  
Ghz States ◽  
Necessary And Sufficient

A Multipartite entangled state has many different kinds of entanglements specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger–Horne–Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For an [Formula: see text] qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into [Formula: see text] parties with [Formula: see text] for arbitrary [Formula: see text] and [Formula: see text]. The criterion covers more than a half of all kinds of partial entanglements for [Formula: see text]-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider [Formula: see text] qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.

An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps

1994 ◽  
Vol 14 (4) ◽  
pp. 621-632 ◽  
Author(s):  
V. Baladi ◽  
D. Ruelle
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Constant Weight ◽  
Piecewise Constant ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Interval Map ◽  
Piecewise Continuous

AbstractWe consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.

scholarly journals Inverse spectral problem for Dirac operators by spectral data

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1065-1077
Author(s):  
Ozge Akcay ◽  
Khanlar Mamedov
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Spectral Problem ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Inverse Spectral Problem ◽  
Sufficient Conditions ◽  
Dirac Operators ◽  
Piecewise Continuous ◽  
Necessary And Sufficient

This work deals with the solution of the inverse problem by spectral data for Dirac operators with piecewise continuous coefficient and spectral parameter contained in boundary condition. The main theorem on necessary and sufficient conditions for the solvability of inverse problem is proved. The algorithm of the reconstruction of potential according to spectral data is given.

scholarly journals Bell-CHSH violation under global unitary operations: Necessary and sufficient conditions

2018 ◽  
Vol 16 (04) ◽  
pp. 1850040 ◽  
Author(s):  
Nirman Ganguly ◽  
Amit Mukherjee ◽  
Arup Roy ◽  
Some Sankar Bhattacharya ◽  
Biswajit Paul ◽  
...  
Keyword(s):  
Hilbert Space ◽  
Comparative Study ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Criterion ◽  
Density Matrices ◽  
Cartan Decomposition ◽  
Chsh Inequality ◽  
Necessary And Sufficient ◽  
Local Filtering

The relation between Bell-CHSH violation and factorization of Hilbert space is considered here. That is, a state which is local in the sense of the Bell-CHSH inequality under a certain factorization of the underlying Hilbert space can be Bell-CHSH nonlocal under a different factorization. While this question has been addressed with respect to separability, the relation of the factorization with Bell-CHSH violation has remained hitherto unexplored. We find here that there is a set containing density matrices, which do not exhibit Bell-CHSH violation under any factorization of the Hilbert space brought about by global unitary operations. Using the Cartan decomposition of [Formula: see text], we characterize the set in terms of a necessary and sufficient criterion based on the spectrum of density matrices. Sufficient conditions are obtained to characterize such density matrices based on their bloch representations. For some classes of density matrices, necessary and sufficient conditions are derived in terms of bloch parameters. Furthermore, an estimation of the volume of such density matrices is achieved in terms of purity. The criterion is applied to some well-known class of states in two qubits. Since both local filtering and global unitary operations influence the Bell-CHSH violation of a state, a comparative study is made between the two operations. The inequivalence of the two operations (in terms of increasing Bell-CHSH violation) is exemplified through their action on some classes of states.

scholarly journals Circle maps and -algebras

2013 ◽  
Vol 35 (2) ◽  
pp. 546-584 ◽  
Author(s):  
THOMAS LUNDSGAARD SCHMIDT ◽  
KLAUS THOMSEN
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Circle Maps ◽  
Piecewise Monotone ◽  
Product Construction ◽  
Algorithmic Method ◽  
Monotone Maps ◽  
Necessary And Sufficient

AbstractWe consider a construction of ${C}^{\ast } $-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the ${C}^{\ast } $-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the $K$-theory groups and turn them into an algorithmic method for Markov maps.

scholarly journals Distributions of Order Patterns of Interval Maps

2013 ◽  
Vol 22 (3) ◽  
pp. 319-341 ◽  
Author(s):  
AARON ABRAMS ◽  
ERIC BABSON ◽  
HENRY LANDAU ◽  
ZEPH LANDAU ◽  
JAMES POMMERSHEIM
Keyword(s):  
Probability Distribution ◽  
Lebesgue Measure ◽  
Compatibility Condition ◽  
Necessary Condition ◽  
Piecewise Monotone ◽  
Interval Maps ◽  
Piecewise Continuous ◽  
Order Pattern ◽  
Exact Description

A permutation σ describing the relative orders of the first n iterates of a point x under a self-map f of the interval I=[0,1] is called an order pattern. For fixed f and n, measuring the points x ∈ I (according to Lebesgue measure) that generate the order pattern σ gives a probability distribution μn(f) on the set of length n permutations. We study the distributions that arise this way for various classes of functions f.Our main results treat the class of measure-preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each n this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general f, apart from an obvious compatibility condition, there is no restriction on the sequence {μn(f)}n=1,2,. . ..In addition, we give a necessary condition for f to have finite exclusion type, that is, for there to be finitely many order patterns that generate all order patterns not realized by f. Using entropy we show that if f is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then f cannot have finite exclusion type. This generalizes results of S. Elizalde.

Topological conjugation of Lorenz maps by β-transformations

1990 ◽  
Vol 107 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Topological Entropy ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Necessary And Sufficient ◽  
Constant Slope ◽  
Lorenz Maps ◽  
Basic Sets ◽  
Topological Conjugation

AbstractNecessary and sufficient conditions for a Lorenz map to be topologically conjugate to a piecewise linear map with constant slope (a β-transformation) are given, first in terms of kneading invariants of the maps and then in terms of the topological entropy restricted to basic sets. The dynamics of β-transformations is also described.

scholarly journals CRITERIA FOR DICHOTOMY OF LINEAR IMPULSIVE DIFFERENTIAL EQUATIONS

1994 ◽  
Vol 25 (2) ◽  
pp. 101-112
Author(s):  
P. S. SIMEONOV ◽  
D. D. BAINOV
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Necessary And Sufficient Conditions ◽  
Piecewise Continuous ◽  
Necessary And Sufficient

In the present paper necessary and sufficient conditions for $(\mu_1, \mu_2)$- dichotomy of linear impulsive differential equations are obtained without imposing conditions of bounded growth on these equations. The apparatus of piecewise continuous Lyapunov's functions is used.

scholarly journals Embedding the symbolic dynamics of Lorenz maps

2014 ◽  
Vol 156 (3) ◽  
pp. 505-519 ◽  
Author(s):  
TONY SAMUEL ◽  
NINA SNIGIREVA ◽  
ANDREW VINCE
Keyword(s):  
Topological Entropy ◽  
Symbolic Dynamics ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Interval Map ◽  
Piecewise Continuous ◽  
Necessary And Sufficient ◽  
Lorenz Maps

AbstractNecessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

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