scholarly journals Ergodic properties of matrix equilibrium states

2017 ◽  
Vol 38 (6) ◽  
pp. 2295-2320 ◽  
Author(s):  
IAN D. MORRIS
Keyword(s):  
Equilibrium State ◽  
Lyapunov Exponents ◽  
Equilibrium States ◽  
Invariant Equilibrium ◽  
Sufficient Condition ◽  
Ergodic Properties ◽  
General Sufficient Condition ◽  
Algebraic Properties ◽  
Bernoulli Measure ◽  
Zero Entropy

Given a finite irreducible set of real$d\times d$matrices$A_{1},\ldots ,A_{M}$and a real parameter$s>0$, there exists a unique shift-invariant equilibrium state on$\{1,\ldots ,M\}^{\mathbb{N}}$associated to$(A_{1},\ldots ,A_{M},s)$. In this paper we characterize the ergodic properties of such equilibrium states in terms of the algebraic properties of the semigroup generated by the associated matrices. We completely characterize when the equilibrium state has zero entropy, when it gives distinct Lyapunov exponents to the natural cocycle generated by$A_{1},\ldots ,A_{M}$, and when it is a Bernoulli measure. We also give a general sufficient condition for the equilibrium state to be mixing, and give an example where the equilibrium state is ergodic but not totally ergodic. Connections with a class of measures investigated by Kusuoka are explored in an appendix.

scholarly journals A necessary and sufficient condition for a matrix equilibrium state to be mixing

2017 ◽  
Vol 39 (8) ◽  
pp. 2223-2234 ◽  
Author(s):  
IAN D. MORRIS
Keyword(s):  
Equilibrium State ◽  
Sufficient Conditions ◽  
Equilibrium States ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hölder Continuous ◽  
Ergodic Properties ◽  
Bernoulli Measure ◽  
Necessary And Sufficient

Since the 1970s there has been a rich theory of equilibrium states over shift spaces associated to Hölder-continuous real-valued potentials. The construction of equilibrium states associated to matrix-valued potentials is much more recent, with a complete description of such equilibrium states being achieved by Feng and Käenmäki [Equilibrium states of the pressure function for products of matrices.Discrete Contin. Dyn. Syst.30(3) (2011), 699–708]. In a recent article [Ergodic properties of matrix equilibrium states.Ergod. Th. & Dynam. Sys.(2017), to appear] the author investigated the ergodic-theoretic properties of these matrix equilibrium states, attempting in particular to give necessary and sufficient conditions for mixing, positive entropy, and the property of being a Bernoulli measure with respect to the natural partition, in terms of the algebraic properties of the semigroup generated by the matrices. Necessary and sufficient conditions were successfully established for the latter two properties, but only a sufficient condition for mixing was given. The purpose of this note is to complete that investigation by giving a necessary and sufficient condition for a matrix equilibrium state to be mixing.

scholarly journals Effects of Diurnal Variations on Tropical Equilibrium States: A Two-Dimensional Cloud-Resolving Modeling Study

2007 ◽  
Vol 64 (2) ◽  
pp. 656-664 ◽  
Author(s):  
Shouting Gao ◽  
Yushu Zhou ◽  
Xiaofan Li
Keyword(s):  
Equilibrium State ◽  
Surface Temperature ◽  
Sea Surface Temperature ◽  
Zenith Angle ◽  
Solar Zenith Angle ◽  
Diurnal Variations ◽  
Equilibrium States ◽  
Sea Surface ◽  
Two Dimensional ◽  
Time Invariant

Abstract Effects of diurnal variations on tropical heat and water vapor equilibrium states are investigated based on hourly data from two-dimensional cloud-resolving simulations. The model is integrated for 40 days and the simulations reach equilibrium states in all experiments. The simulation with a time-invariant solar zenith angle produces a colder and drier equilibrium state than does the simulation with a diurnally varied solar zenith angle. The simulation with a diurnally varied sea surface temperature generates a colder equilibrium state than does the simulation with a time-invariant sea surface temperature. Mass-weighted mean temperature and precipitable water budgets are analyzed to explain the thermodynamic differences. The simulation with the time-invariant solar zenith angle produces less solar heating, more condensation, and consumes more moisture than the simulation with the diurnally varied solar zenith angle. The simulation with the diurnally varied sea surface temperature produces a colder temperature through less latent heating and more IR cooling than the simulation with the time-invariant sea surface temperature.

A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have?

2009 ◽  
Vol 18 (5) ◽  
pp. 691-705 ◽  
Author(s):  
GYÖRGY ELEKES ◽  
MIKLÓS SIMONOVITS ◽  
ENDRE SZABÓ
Keyword(s):  
Parameter Family ◽  
Sufficient Condition ◽  
Simple Application ◽  
Quadratic Number ◽  
General Sufficient Condition ◽  
Family Of Curves ◽  
Unit Circles ◽  
Triple Points ◽  
Straight Lines ◽  
Special Case

We give a very general sufficient condition for a one-parameter family of curves not to have n members with ‘too many’ (i.e., a near-quadratic number of) triple points of intersections. As a special case, a combinatorial distinction between straight lines and unit circles will be shown. (Actually, this is more than just a simple application; originally this motivated our results.)

A sufficient condition for instability of BGK type waves in both unmagnetized and magnetized plasmas

1977 ◽  
Vol 17 (1) ◽  
pp. 57-68 ◽  
Author(s):  
E. Infeld ◽  
G. Rowlands
Keyword(s):  
Magnetic Field ◽  
Linear Theory ◽  
General Problem ◽  
Uniform Magnetic Field ◽  
Magnetized Plasmas ◽  
Sufficient Condition ◽  
General Sufficient Condition ◽  
Special Cases

This paper investigates the general problem of stability of Bernstein—Greene— Kruskal type waves. By investigating perturbations perpendicular to the wave, we obtain a general sufficient condition for instability. This is then extended to the case of magnetized plasmas with a uniform magnetic field in the direction of the BGK wave. New perturbed modes, having no counterpart in linear theory, are also found. Various special cases are considered and previous, more particular results confirmed.

scholarly journals Equilibrium states for piecewise monotonic transformations

1982 ◽  
Vol 2 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Franz Hofbauer ◽  
Gerhard Keller
Keyword(s):  
Periodic Orbits ◽  
Bounded Variation ◽  
Critical Points ◽  
Equilibrium States ◽  
Absolutely Continuous ◽  
Specification Property ◽  
Ergodic Properties ◽  
Piecewise Monotonic

AbstractWe show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:(i) sup φ — inf φ <htop(T) and φ is of bounded variation.(ii) φ satisfies a variation condition and T has a local specification property.(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

A sufficient condition for zero entropy

2014 ◽  
Vol 175 (3) ◽  
pp. 323-332 ◽  
Author(s):  
A. Arbieto ◽  
C. Morales
Keyword(s):  
Sufficient Condition ◽  
Zero Entropy
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