Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property

@article{Garibaldi2020DynamicalOT,
  title={Dynamical obstruction to the existence of continuous sub-actions for interval maps with regularly varying property},
  author={Eduardo Garibaldi and Irene Inoquio-Renteria},
  journal={Discrete \& Continuous Dynamical Systems - A},
  year={2020}
}
In ergodic optimization theory, the existence of sub-actions is an important tool in the study of the so-called optimizing measures. For transformations with regularly varying property, we highlight a class of moduli of continuity which is not compatible with the existence of continuous sub-actions. Our result relies fundamentally on the local behavior of the dynamics near a fixed point and applies to interval maps that are expanding outside an indifferent fixed point, including Manneville… 
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References

SHOWING 1-10 OF 28 REFERENCES
Ergodic Optimization
Let f be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic f -average is as…
The Mañé–Conze–Guivarc’h lemma for intermittent maps of the circle
  • I. Morris
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2009
Abstract We study the existence of solutions g to the functional inequality f≤g ∘T−g+β, where f is a prescribed continuous function, T is a weakly expanding transformation of the circle having an…
Slowly mixing systems and intermittency maps
  • M. Holland
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2004
We consider families of one-dimensional maps on the circle which mix at sub-polynomial rates. Such maps will have an indifferent fixed point, and we show that the rate of mixing of these maps is…
Sub-actions for Anosov diffeomorphisms
We show a positive Livciz type theorem for C2 Anosov diffeomorphisms f on a compact boundaryless manifold M and Hölder observables A. Given A : M → R, α-Hölder, we show there exist V : M → R,…
Cohomology classes of dynamically non-negative Ck functions
Abstract.Let T:x↦2x (mod 1) be the doubling map of the circle ?=ℝ/ℤ. We construct a trigonometric polynomial f:?→ℝ with the following property: ∫f  dμ≥0 for every T-invariant probability measure μ,…
Cohomology and subcohomology problems for expansive, non Anosov geodesic flows
We show that there are examples of expansive, non-Anosov geodesic flows of compact surfaces with non-positive curvature, where the Livsic Theorem holds in its classical (continuous, Holder)…
On calibrated and separating sub-actions
AbstractWe consider a one-sided transitive subshift of finite type σ: Σ → Σ and a Hölder observable A. In the ergodic optimization model, one is interested in properties of A-minimizing probability…
A sufficient condition for the subordination principle in ergodic optimization
Let T : X → X be a continuous surjection of a topological space, and let f : X → ℝ be upper semi‐continuous. We wish to identify those T‐invariant measures μ which maximize ∫ f dμ. We call such…
Sub-actions for weakly hyperbolic one-dimensional systems
Let f be an expanding map of degree 2 in the unitary interval [0,1], with an indifferent fixed point at x = 0 (i. e. (0) = 1), increasing, surjective and C 1 in each injective branch [0, c ] and ( c…
Ergodic optimization in dynamical systems
  • O. Jenkinson
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2019
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…
...
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