Ergodic optimization in dynamical systems

Jenkinson, Oliver

doi:10.1017/etds.2017.142

Cited by 80 publications

(76 citation statements)

References 156 publications

(283 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In traditional ergodic optimization, that is, the optimization of Birkhoff averages (see [Je1,Je2,Gar]), a maximizing set is a closed subset such that an invariant probability is maximizing if and only if its support lies on this subset. The existence of such sets is guaranteed in any context where a Mañé Lemma holds.…”

Section: Mather Setsmentioning

confidence: 99%

Extremal norms for fiber-bunched cocycles

Bochi¹,

Garibaldi²

2019

Journal De L’École Polytechnique — Mathématiques

View full text Add to dashboard Cite

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.

show abstract

Section: Mather Setsmentioning

confidence: 99%

Extremal norms for fiber-bunched cocycles

Bochi¹,

Garibaldi²

2019

Journal De L’École Polytechnique — Mathématiques

View full text Add to dashboard Cite

show abstract

“…, p − 1} and consider the gap X a 0 ···a k−1 i (h n−1 ) of order k. Let x ∈ X a 0 ···a k−1 i (h n−1 ). From (10), (15) |f 0 (x) − f (x)| ≤ f 0 (X a 0 ···a k−1 i (h n−1 )) ≤ ε 2 .…”

Section: Approximation By Maps With Locally Constant Derivativesmentioning

confidence: 99%

“…|Df 0 (x) − Df (x)| ≤ ||Df 0 (x)| − τ | + |τ − |Df (x)|| ≤ ε 4From (12),(13),(14),(15),(16), f 0 − f C 1 ≤ ε follows. By Lemma 2.2, (E2) holds for f .…”

mentioning

confidence: 92%

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

Shinoda¹,

Takahasi²

2019

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

For a non-generic, yet dense subset of C 1 expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C 1 perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

show abstract

“…We say a measure µ is a ground state of the potential ϕ if µ is the weak * limit of a sequence of equilibrium states µ tn ∈ ES(t n ϕ) for some sequence t n → ∞. It follows that every ground state µ of ϕ is a maximizing measure, that is µ maximizes the integral ϕdν among all invariant probability measures (see [16]). In the presence of a unique ground state (i.e.…”

mentioning

confidence: 99%

Ground states and zero-temperature measures at the boundary of rotation sets

Kucherenko¹,

Wolf²

2017

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We consider a continuous dynamical system f : X → X on a compact metric space X equipped with an m-dimensional continuous potential Φ = (φ1, · · · , φm) : X → R m . We study the set of ground states GS(α) of the potential α · Φ as a function of the direction vector α ∈ S m−1 . We show that the structure of the ground state sets is naturally related to the geometry of the generalized rotation set of Φ. In particular, for each α the set of rotation vectors of GS(α) forms a non-empty, compact and connected subset of a face Fα(Φ) of the rotation set associated with α. Moreover, every ground state maximizes entropy among all invariant measures with rotation vectors in Fα(Φ). We further establish the occurrence of several quite unexpected phenomena. Namely, we construct for any m ∈ N examples with an exposed boundary point (i.e. Fα(Φ) being a singleton) without a unique ground state. Further, we establish the possibility of a line segment face Fα(Φ) with a unique but non-ergodic ground state. Finally, we establish the possibility that the set of rotation vectors of GS(α) is a non-trivial line segment.

show abstract

Ergodic optimization in dynamical systems

Cited by 80 publications

References 156 publications

Extremal norms for fiber-bunched cocycles

Extremal norms for fiber-bunched cocycles

Lyapunov optimization for non-generic one-dimensional expanding Markov maps

Ground states and zero-temperature measures at the boundary of rotation sets

Contact Info

Product

Resources

About