| 1. |
- Abadie, F., et al.
(författare)
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Partial C*-dynamics and Rokhlin dimension
- 2022
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Ingår i: Ergodic Theory and Dynamical Systems. - 0143-3857 .- 1469-4417. ; 42, s. 2991-3024
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Tidskriftsartikel (refereegranskat)
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| 2. |
- Aspenberg, Magnus, et al.
(författare)
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Hausdorff dimension of escaping sets of meromorphic functions II
- 2023
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 43:5, s. 1471-1491
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Tidskriftsartikel (refereegranskat)abstract
- A function which is transcendental and meromorphic in the plane has at least two singular values. On the one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either or . On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in (cf. [M. Aspenberg and W. Cui. Hausdorff dimension of escaping sets of meromorphic functions. Trans. Amer. Math. Soc. 374(9) (2021), 6145-6178]). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than .
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| 3. |
- Benedicks, Michael, Professor, et al.
(författare)
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Expansion properties of double standard maps
- 2023
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 43:8, s. 2549-2588
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Tidskriftsartikel (refereegranskat)abstract
- For the family of double standard maps we investigate the structure of the space of parameters a when and when. In the first case the maps have a critical point, but for a set of parameters of positive Lebesgue measure there is an invariant absolutely continuous measure for. In the second case there is an open non-empty set of parameters for which the map is expanding. We show that as, the set accumulates on many points of in a regular way from the measure point of view.
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| 4. |
- Bjerklöv, Kristian, Docent
(författare)
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On some generalizations of skew-shifts on T-2
- 2019
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Ingår i: Ergodic Theory and Dynamical Systems. - : CAMBRIDGE UNIV PRESS. - 0143-3857 .- 1469-4417. ; 39, s. 19-61
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we investigate maps of the two-torus T-2 of the form T (x, y) = (x + omega, g(x) + f (y)) for Diophantine omega is an element of T and for a class of maps f, g : T -> T, where each g is strictly monotone and of degree 2 and each f is an orientation-preserving circle homeomorphism. For our class of f and g, we show that T is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two T-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in T-2. Only a low-regularity assumption (Lipschitz) is needed on the maps f and g, and the results are robust with respect to Lipschitz-small perturbations of f and g.
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| 5. |
- Bjerklöv, Kristian
(författare)
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Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrodinger equations
- 2005
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 25, s. 1015-1045
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Tidskriftsartikel (refereegranskat)abstract
- We study the discrete quasi-periodic Schrodinger equation -(u(n+1) + u(n-1)) + lambda V(theta + n omega)u(n) = Eu-n with a non-constant C-1 potential function V : T -> R. We prove that for sufficiently large k there is a set Omega subset of T of frequencies omega, whose measure tends to 1 as lambda -> infinity, with the following property. For each w e Q there is a 'large' (in measure) set of energies E, all lying in the spectrum of the associated Schrodinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrodinger cocycle is minimal but not ergodic.
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| 6. |
- Björklund, Michael, 1981
(författare)
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Product set phenomena for measured groups
- 2018
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 38:8, s. 2913-2941
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Tidskriftsartikel (refereegranskat)abstract
- We strengthen and extend in this paper some recent results by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg on piecewise syndeticity of product sets in countable amenable groups to general countable measured groups. We also address several fundamental differences between the behavior of products of 'large' sets in Liouville and non-Liouville measured groups. As a (very) special case of our main results, we show that if G is a free group of finite rank, and A and B are 'spherically large' subsets of G, then there exists a finite set F subset of G such that AFB is thick. The position of the set F is curious, but seems to be necessary; in fact, we can produce left thick sets A, B subset of G such that B is 'spherically large', but AB is not piecewise syndetic. On the other hand, if A is spherically large, then AA(-1) is always piecewise syndetic and piecewise left syndetic. However, contrary to what happens for amenable groups, AA(-1) may fail to be syndetic. The same phenomena occur for many other (even amenable, but non-Liouville) measured groups. Our proofs are based on some ergodic-theoretical results concerning stationary actions which should be of independent interest.
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| 7. |
- Bylund, Mats
(författare)
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Critical recurrence in the real quadratic family
- 2023
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 43:10, s. 3255-3287
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Tidskriftsartikel (refereegranskat)abstract
- We study recurrence in the real quadratic family and give a sufficient condition on the recurrence rate of the critical orbit such that, for almost every non-regular parameter a, the set of n such that Fn(0;a) is infinite. In particular, when, this extends an earlier result by Avila and Moreira [Statistical properties of unimodal maps: the quadratic family. Ann. of Math. (2) 161(2) (2005), 831-881].
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| 8. |
- Damjanovic, Danijela, et al.
(författare)
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Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions
- 2020
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Ingår i: Ergodic Theory and Dynamical Systems. - : CAMBRIDGE UNIV PRESS. - 0143-3857 .- 1469-4417. ; 40:1, s. 117-141
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Tidskriftsartikel (refereegranskat)abstract
- We prove that every smooth diffeomorphism group valued cocycle over certain Z(k) Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan Z(k) (k >= 3) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over Z(k) actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.
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| 9. |
- Damjanović, Danijela, et al.
(författare)
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Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds
- 2021
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; , s. 1-41
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove a perturbative result for a class of actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.
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| 10. |
- De la Llave, Rafael, et al.
(författare)
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Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation
- 2022
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 42:3, s. 1166-1187
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Tidskriftsartikel (refereegranskat)abstract
- Consider an analytic Hamiltonian system near its analytic invariant torus T-0 carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at T-0 is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation-not just a formal power series-bringing the Hamiltonian into its Birkhoff normal form.
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