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Journal of Modern Dynamics

 2021 , Volume 17

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On mixing and sparse ergodic theorems
Asaf Katz
2021, 17: 1-32 doi: 10.3934/jmd.2021001 +[Abstract](161) +[HTML](74) +[PDF](294.76KB)
Abstract:

We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allow us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis, Shah, and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent of the spectral gap.

Dynamics of 2-interval piecewise affine maps and Hecke-Mahler series
Michel Laurent and Arnaldo Nogueira
2021, 17: 33-63 doi: 10.3934/jmd.2021002 +[Abstract](110) +[HTML](50) +[PDF](369.73KB)
Abstract:

Let \begin{document}$ f : [0,1)\rightarrow [0,1) $\end{document} be a \begin{document}$ 2 $\end{document}-interval piecewise affine increasing map which is injective but not surjective. Such a map \begin{document}$ f $\end{document} has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of \begin{document}$ f $\end{document} thanks to two specific functions \begin{document}$ {\boldsymbol{\delta}} $\end{document} and \begin{document}$ \phi $\end{document} depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of \begin{document}$ f $\end{document} is rational, whenever the three parameters are all algebraic numbers, extending thus the main result of [16] dealing with the particular case of \begin{document}$ 2 $\end{document}-interval piecewise affine contractions with constant slope.

Local Lyapunov spectrum rigidity of nilmanifold automorphisms
Jonathan DeWitt
2021, 17: 65-109 doi: 10.3934/jmd.2021003 +[Abstract](64) +[HTML](24) +[PDF](715.77KB)
Abstract:

We study the regularity of a conjugacy between an Anosov automorphism \begin{document}$ L $\end{document} of a nilmanifold \begin{document}$ N/\Gamma $\end{document} and a volume-preserving, \begin{document}$ C^1 $\end{document}-small perturbation \begin{document}$ f $\end{document}. We say that \begin{document}$ L $\end{document} is locally Lyapunov spectrum rigid if this conjugacy is \begin{document}$ C^{1+} $\end{document} whenever \begin{document}$ f $\end{document} is \begin{document}$ C^{1+} $\end{document} and has the same volume Lyapunov spectrum as \begin{document}$ L $\end{document}. For \begin{document}$ L $\end{document} with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to \begin{document}$ L $\end{document} satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.

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