scholarly journals Equilibrium states for a class of skew products

2019 ◽  
Vol 40 (11) ◽  
pp. 3030-3050
Author(s):  
MARIA CARVALHO ◽  
SEBASTIÁN A. PÉREZ
Keyword(s):  
Existence And Uniqueness ◽  
Global Analysis ◽  
American Mathematical Society ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Axiom A ◽  
Skew Products ◽  
Pure Mathematics ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We consider skew products on $M\times \mathbb{T}^{2}$, where $M$ is the two-sphere or the two-torus, which are partially hyperbolic and semi-conjugate to an Axiom A diffeomorphism. This class of dynamics includes the open sets of $\unicode[STIX]{x1D6FA}$-non-stable systems introduced by Abraham and Smale [Non-genericity of Ł-stability. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV (Berkeley 1968)). American Mathematical Society, Providence, RI, 1970, pp. 5–8.] and Shub [Topological Transitive Diffeomorphisms in$T^{4}$ (Lecture Notes in Mathematics, 206). Springer, Berlin, 1971, pp. 39–40]. We present sufficient conditions, both on the skew products and the potentials, for the existence and uniqueness of equilibrium states, and discuss their statistical stability.

scholarly journals On equilibrium states for partially hyperbolic horseshoes

2016 ◽  
Vol 38 (1) ◽  
pp. 301-335 ◽  
Author(s):  
I. RIOS ◽  
J. SIQUEIRA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Three Dimensional ◽  
Small Variation ◽  
Equilibrium States ◽  
Dimensional System ◽  
Hölder Continuous ◽  
The Family ◽  
Partially Hyperbolic ◽  
Uniqueness Of Equilibrium

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

scholarly journals Extendability of Equilibria of Nematic Polymers

2008 ◽  
Vol 2008 ◽  
pp. 1-10
Author(s):  
Hongyun Wang ◽  
Hong Zhou
Keyword(s):  
Nonlinear System ◽  
Existence And Uniqueness ◽  
Jacobian Matrix ◽  
Intersection Point ◽  
Equilibrium States ◽  
Intermolecular Potential ◽  
Small Perturbations ◽  
Nematic Polymers ◽  
Folding Point ◽  
Uniqueness Of Equilibrium

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

scholarly journals Equilibrium states for Mañé diffeomorphisms

2018 ◽  
Vol 39 (9) ◽  
pp. 2433-2455 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
TODD FISHER ◽  
DANIEL J. THOMPSON
Keyword(s):  
Large Deviations ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Potential Functions ◽  
Equilibrium States ◽  
Unique Equilibrium ◽  
The Family ◽  
Natural Classes ◽  
Uniqueness Of Equilibrium

We study thermodynamic formalism for the family of robustly transitive diffeomorphisms introduced by Mañé, establishing existence and uniqueness of equilibrium states for natural classes of potential functions. In particular, we characterize the Sinaĭ–Ruelle–Bowen measures for these diffeomorphisms as unique equilibrium states for a suitable geometric potential. We also obtain large deviations and multifractal results for the unique equilibrium states produced by the main theorem.

scholarly journals Existence and uniqueness of equilibrium states for some spin and continuum systems

1973 ◽  
Vol 32 (2) ◽  
pp. 153-165 ◽  
Author(s):  
M. Cassandro ◽  
G. Gallavotti ◽  
J. L. Lebowitz ◽  
J. L. Monroe
Keyword(s):  
Existence And Uniqueness ◽  
Equilibrium States ◽  
Uniqueness Of Equilibrium ◽  
Continuum Systems

Thermodynamic formalism for a family of non-uniformly hyperbolic horseshoes and the unstable Jacobian

2010 ◽  
Vol 31 (2) ◽  
pp. 423-447 ◽  
Author(s):  
RENAUD LEPLAIDEUR
Keyword(s):  
Lyapunov Exponent ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Fractal Dimensions ◽  
Equilibrium States ◽  
Real Analytic ◽  
Homoclinic Tangencies ◽  
Uniqueness Of Equilibrium

AbstractIn this article we prove the existence and uniqueness of equilibrium states for the potential $\phi _{t}= -t\logju \ (t\in \R )$ and the class of non-uniformly hyperbolic horseshoes which was introduced in Rios [Unfolding homoclinic tangencies inside horseshoes: hyperbolicity, fractal dimensions and persistent tangencies. Nonlinearity14 (2001), 431–462]. We show that the pressure t↦𝒫(t) for −tlog Ju is real-analytic on $\R $. We give the exact equations of the two asymptotes to the graph of 𝒫(t) at ±∞ and we prove that these non-uniformly hyperbolic horseshoes do not have measures which minimize the unstable Lyapunov exponent.

scholarly journals Existence of Coupled Best Proximity Points of p-Cyclic Contractions

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 39
Author(s):  
Miroslav Hristov ◽  
Atanas Ilchev ◽  
Diana Nedelcheva ◽  
Boyan Zlatanov
Keyword(s):  
Wide Class ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Best Proximity Points ◽  
Cyclic Contractions ◽  
Ordered Pairs

We generalize the notion of coupled fixed (or best proximity) points for cyclic ordered pairs of maps to p-cyclic ordered pairs of maps. We find sufficient conditions for the existence and uniqueness of the coupled fixed (or best proximity) points. We illustrate the results with an example that covers a wide class of maps.

On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty

2020 ◽  
Vol 20 (2) ◽  
Author(s):  
Stefanos Leonardos ◽  
Costis Melolidakis
Keyword(s):  
Failure Rate ◽  
Demand Uncertainty ◽  
Sufficient Conditions ◽  
Cournot Competition ◽  
Uncertain Demand ◽  
Cournot Model ◽  
Equilibrium Uniqueness ◽  
Residual Demand ◽  
Uniqueness Of Equilibrium ◽  
Additional Assumptions

AbstractWe revisit the linear Cournot model with uncertain demand that is studied in Lagerlöf (2006. “Equilibrium Uniqueness in a Cournot Model with Demand Uncertainty.” The B.E. Journal of Theoretical Economics 6, no. 1. (Topics), Article 19: 1–6.) and provide sufficient conditions for equilibrium uniqueness that complement the existing results. We show that if the distribution of the demand intercept has the decreasing mean residual demand (DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibrium is guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of output without additional assumptions on the existence or the shape of the density of the demand intercept and, hence, answers in the affirmative the conjecture of Lagerlöf (2006. “Equilibrium Uniqueness in a Cournot Model with Demand Uncertainty.” The B.E. Journal of Theoretical Economics 6, no. 1. (Topics), Article 19: 1–6.) that such conditions may not be necessary.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

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