Random Products of Standard Maps

Carrasco, Pablo D.

doi:10.1007/s00220-020-03765-6

Cited by 4 publications

(5 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this part we will establish the remaining part of Theorem A and prove that for t sufficiently large, the map f = f t is NUH. The arguments are inspired in the abstract methods used in [CS22] (see also [Car20]), but with the additional difficulty of having to work in the natural extension instead of directly in the manifold. We rely on Proposition 2.2.…”

Section: Non-uniform Hyperbolicitymentioning

confidence: 99%

Non-uniformly hyperbolic endomorphisms

Andersson¹,

Carrasco²,

Saghin³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We show the existence of large C 1 open sets of area preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the C 1 topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.

show abstract

Section: Non-uniform Hyperbolicitymentioning

confidence: 99%

Non-uniformly hyperbolic endomorphisms

Andersson¹,

Carrasco²,

Saghin³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This means that we gain some expansion in this good region, however there is a difficulty in the fact that the good region is not invariant, and once an orbit goes out of the good region we may loose control on the expansion which we gained. In order to deal with this problem we use ideas from [BC14], [Car20]. We consider a family of unit vectorfields inside which are Lipschitz along .…”

Section: Ideas Of the Proofsmentioning

confidence: 99%

“…In pursue of versatility, the result is written in some general setting. The method of the proof is based in [BC14] and its subsequent refinement [Car20].…”

Section: Bounding the Lyapunov Exponents From Belowmentioning

confidence: 99%

See 1 more Smart Citation

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Carrasco¹,

Saghin²

2021

Preprint

View full text Add to dashboard Cite

Bochi-Katok-Rodriguez Hertz proposed in [BKH] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy entropy larger than the entropy of the linear part.These results rely in a new type of deformation method for conservative systems having some invariant directions. We also comment on the C 2 robustness of the examples.

show abstract

“…However, they restrict the fiber map

f

to one of some particular classes to ensure contraction or hyperbolic properties (exact or averaged) of the vertical fiber. Skew‐products with nonuniform hyperbolicity can still be studied but in a more qualitative sense [5, 12]. In contrast, our results make only mild regularity assumptions on

f

, but require that

g

is uniformly expanding with large minimal expansion.…”

Section: Introductionmentioning

confidence: 99%

Random‐like properties of chaotic forcing

Giulietti

Marmi

Tanzi

2022

Journal of London Math Soc

View full text Add to dashboard Cite

We prove that skew systems with a sufficiently expanding base have approximate exponential decay of correlations, meaning that the exponential rate is observed modulo an error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. However, the error in the approximation goes to zero when the expansion of the base tends to infinity. The result can be applied beyond the original setup when combined with acceleration or conjugation arguments, as our examples show.

show abstract

Random Products of Standard Maps

Cited by 4 publications

References 37 publications

Non-uniformly hyperbolic endomorphisms

Non-uniformly hyperbolic endomorphisms

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Random‐like properties of chaotic forcing

Contact Info

Product

Resources

About