Problem 8

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Non Axiom A examples. Newhouse, Abraham-Smale, Simon, Lorenz, billiards.

  1. Axiomatic description
  2. Statistical properties
  3. For all $\epsilon$, there exists a horseshoe $X_\epsilon$ inside with $ h(f|X_\epsilon) \geq h(f) - \epsilon$
  4. Statistical properties of Lorenz in particular
  5. Any specification type property

Tags

Tags: 
Hyperbolic
Entropy

Comments

Submitted by Ledrappier on Thu, 11/10/2016 - 11:58

c. proved by Katok ([1]) in dimension 2 and more generally when there is no zero Lyapunov exponents for measures with entropies arbitrarily close to $h(f)$.

References

  1. [katok1980lyapunov] Katok A.  1980.  Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publications Mathématiques de l'IHÉS. 51:137–173.

Submitted by BrianMarcus on Thu, 11/10/2016 - 11:58

(For part e.) While perhaps not related so much to the kinds of examples Rufus had in mind here, various kinds of specification properties have been developed in connection with uniqueness of measures of maximal entropy, e.g., Climenhaga-Thompson [1], Kulczycki, Kwietniak, and Oprocha [2], and Pavlov [3]; see also problem (32)

References

  1. [climenhaga2012intrinsic] Climenhaga V, Thompson DJ.  2012.  Intrinsic ergodicity beyond specification: $\beta$-shifts, S-gap shifts, and their factors. Israel Journal of Mathematics. 192:785–817.
  2. [kulczycki2013almost] Kulczycki M, Kwietniak D, Oprocha P.  2013.  On almost specification and average shadowing properties. arXiv preprint arXiv:1307.0120.
  3. [pavlov2016intrinsic] Pavlov R.  2016.  On intrinsic ergodicity and weakenings of the specification property. Advances in Mathematics. 295:250–270.

Submitted by Vaughn Climenhaga on Fri, 07/14/2017 - 22:10

Regarding (weaker) specification properties: The approach in [C.-Thompson 2012] has recently been extended (by C.--Fisher--Thompson) to some smooth systems, including examples due to Mañé [arXiv:1703.05722], Bonatti-Viana [arXiv:1505.06371], as well as (by Burns--C.--Fisher--Thompson) to geodesic flows in nonpositive curvature [arxiv.org:1703.10878].  To my eye, the Mañé and Bonatti-Viana examples share some significant features with the Abraham-Smale and Simon examples.  The main thing driving the approach in the preprints cited above is that there is some region B such that the specification property holds for orbit segments that spend "enough" time outside of B, so B in some sense represents the "obstructions" to specification.  Then if entropy of B is smaller than total entropy, one gets a unique MME (modulo some other technicalities), and a similar result holds for pressure and equilibrium states.  It seems reasonable to me to expect that a similar game could be played with the Abraham-Smale and Simon examples, but I don't think anyone's looked at this yet.

Submitted by Ledrappier on Thu, 11/10/2016 - 11:58

There have been a lot of activities around all these questions, that would deserve a survey by themselves.

Submitted by michael.jakobson on Wed, 07/12/2017 - 04:39

Related question:

SRB measures for Henon maps which are not small perturbations of 1-d maps.

In particular computer assisted estimates of the measure of SRB parameters near to Henon values a=1.4, b=0.3.

Related question.

Computer assisted estimates of the measure of SRB parameters in the quadratic family x->ax(1-x) for parameters

not too close to a=4.

Submitted by Vaughn Climenhaga on Fri, 07/14/2017 - 19:03

For (d) there is the recent paper by Araujo and Melbourne [Exponential decay of correlations for nonuniformly hyperbolic flows with a $C^{1+\alpha}$ stable foliation, including the classical Lorenz attractor. Ann. Henri Poincaré 17 (2016), no. 11, 2975–3004]

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