scholarly journals Ergodic optimization in dynamical systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2593-2618 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Model Problem ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Ergodic Optimization ◽  
Maximizing Measure ◽  
Time Average

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called$f$-maximizing if the time average of the real-valued function$f$along the orbit is larger than along all other orbits, and an invariant probability measure is called$f$-maximizing if it gives$f$a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps

1998 ◽  
Vol 18 (3) ◽  
pp. 555-565 ◽  
Author(s):  
HENK BRUIN
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Topological Invariant ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant ◽  
Critical Order

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

scholarly journals Measure rigidity for algebraic bipermutative cellular automata

2007 ◽  
Vol 27 (6) ◽  
pp. 1965-1990 ◽  
Author(s):  
MATHIEU SABLIK
Keyword(s):  
Cellular Automata ◽  
Probability Measure ◽  
Cellular Automaton ◽  
Compact Group ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Invariant Subgroup ◽  
Shift Map ◽  
Bernoulli Measure ◽  
Measure Rigidity

AbstractLet $({\mathcal {A}^{\mathbb {Z}}} ,F)$ be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the $ {\mathbb {N}} \times {\mathbb {Z}} $-action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group $ {\mathcal {A}^{\mathbb {Z}}} $. This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on $ {\mathcal {A}^{\mathbb {N}}} $.

scholarly journals Multifractal rigidity for piecewise linear Markov maps

2016 ◽  
Vol 17 (01) ◽  
pp. 1750006
Author(s):  
Katsukuni Nakagawa
Keyword(s):  
Piecewise Linear ◽  
Thermodynamic Formalism ◽  
Temperature Limit ◽  
Parameter Family ◽  
Zero Temperature ◽  
Markov Measures ◽  
Multifractal Rigidity

We give an answer to the multifractal rigidity problem presented by Barreira, Pesin and Schmeling for the dimension spectra of Markov measures on the repellers of piecewise linear Markov maps with two branches. Thermodynamic formalism provides us with a one-parameter family of measures. Zero-temperature limit measures of this family and the concept of nondegeneracy of spectra play important roles.

scholarly journals Ergodic and thermodynamic games

2016 ◽  
Vol 16 (02) ◽  
pp. 1660008 ◽  
Author(s):  
Rafael Rigão Souza
Keyword(s):  
Nash Equilibrium ◽  
Cooperative Games ◽  
Invariant Measures ◽  
Equilibrium Points ◽  
Thermodynamic Formalism ◽  
Usual Sense ◽  
Compact Sets ◽  
Nash Equilibrium Points ◽  
Ergodic Optimization ◽  
Non Cooperative Games

Let [Formula: see text] and [Formula: see text] be compact sets, and [Formula: see text], [Formula: see text] be continuous maps. Let [Formula: see text] where [Formula: see text] is [Formula: see text]-invariant and [Formula: see text] is [Formula: see text]-invariant, be payoff functions for a game (in the usual sense of game theory) between players that have the set of invariant measures for [Formula: see text] (player 1) and [Formula: see text] (player 2) as possible strategies. Our goal here is to establish the notion of Nash equilibrium for the game defined by these payoffs and strategies. The main tools come from ergodic optimization (as we are optimizing over the set of invariant measures) and thermodynamic formalism (when we add to the integrals above the entropy of measures in order to define a second case to be explored). Both cases are ergodic versions of non-cooperative games. We show the existence of Nash equilibrium points with two independent arguments. One of the arguments deals with the case with entropy, and uses only tools of thermodynamical formalism, while the other, that works in the case without entropy but can be adapted to deal with both cases, uses the Kakutani fixed point. We also present examples and briefly discuss uniqueness (or lack of uniqueness). In the end, we present a different example where players are allowed to collaborate. This final example shows connections between cooperative games and ergodic transport.

Invariant measures of groups of homeomorphisms and Auslander's conjecture

1995 ◽  
Vol 15 (6) ◽  
pp. 1075-1089 ◽  
Author(s):  
Shmuel Friedland
Keyword(s):  
Probability Measure ◽  
Projective Space ◽  
Complex Projective Space ◽  
Hausdorff Space ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Group Of Homeomorphisms ◽  
Complex Projective

AbstractWe give sufficient conditions for a group of homeomorphisms of a compact Hausdorff space to have an invariant probability measure. For a complex projective space CPn we give a necessary condition for a subgroup of Aut(CPn) to have an invariant probability measure. We discuss two approaches to Auslander's conjecture.

scholarly journals Dissipation of Quantum Turbulence in the Zero Temperature Limit

2007 ◽  
Vol 99 (26) ◽  
Author(s):  
P. M. Walmsley ◽  
A. I. Golov ◽  
H. E. Hall ◽  
A. A. Levchenko ◽  
W. F. Vinen
Keyword(s):  
Quantum Turbulence ◽  
Temperature Limit ◽  
Zero Temperature

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

2014 ◽  
Vol 36 (1) ◽  
pp. 215-255 ◽  
Author(s):  
SAMUEL SENTI ◽  
HIROKI TAKAHASI
Keyword(s):  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Statistical Properties ◽  
Limit Set ◽  
Bifurcation Parameter ◽  
Large Interval ◽  
Equilibrium Measures ◽  
Uniform Hyperbolicity ◽  
Unstable Direction

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential$-t\log J^{u}$, where$t\in \mathbb{R}$is in a certain large interval and$J^{u}$denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

scholarly journals Strong magnon–photon coupling with chip-integrated YIG in the zero-temperature limit

2021 ◽  
Vol 119 (3) ◽  
pp. 033502
Author(s):  
Paul G. Baity ◽  
Dmytro A. Bozhko ◽  
Rair Macêdo ◽  
William Smith ◽  
Rory C. Holland ◽  
...  
Keyword(s):  
Temperature Limit ◽  
Zero Temperature ◽  
Photon Coupling
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