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… 
1 Citations

Figures from this paper

A Ruelle-Perron-Frobenius theorem for expanding circle maps with an indifferent fixed point
In this note, we establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose continuity

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
...
...