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Register a new userUniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps
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We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one dimensional torus translation, these estimates are nearly optimal.
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The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages begin{equation*} frac 1 Nsum_{n=0}^{N-1}f_1(T^nx)cdots f_d(T^{dn}x), quad Nto infty, end{equation*} almost surely converge.
For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on the entropy and Lyapunov spectra.
We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett-Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show that the spectrum varies analytically in parts of its domain. We apply our results to show that the Birkhoff spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable differences with the uniformly hyperbolic setting. We also obtain results describing the Birkhoff spectrum of suspension flows. Examples involving continued fractions are also given.
In this article, we consider the weighted ergodic optimization problem of a class of dynamical systems $T:Xto X$ where $X$ is a compact metric space and $T$ is Lipschitz continuous. We show that once $T:Xto X$ satisfies both the {em Anosov shadowing property }({bf ASP}) and the {em Ma~ne-Conze-Guivarch-Bousch property }({bf MCGBP}), the minimizing measures of generic Holder observables are unique and supported on a periodic orbit. Moreover, if $T:Xto X$ is a subsystem of a dynamical system $f:Mto M$ (i.e. $Xsubset M$ and $f|_X=T$) where $M$ is a compact smooth manifold, the above conclusion holds for $C^1$ observables. Note that a broad class of classical dynamical systems satisfies both ASP and MCGBP, which includes {em Axiom A attractors, Anosov diffeomorphisms }and {em uniformly expanding maps}. Therefore, the open problem proposed by Yuan and Hunt in cite{YH} for $C^1$-observables is solved consequentially.
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