American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

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Journal of Modern Dynamics

2021 , Volume 17

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2021, 17: 1-32 doi: 10.3934/jmd.2021001 +[Abstract](376) +[HTML](173) +[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.

2021, 17: 33-63 doi: 10.3934/jmd.2021002 +[Abstract](278) +[HTML](134) +[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.

2021, 17: 65-109 doi: 10.3934/jmd.2021003 +[Abstract](260) +[HTML](120) +[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.

2021, 17: 111-143 doi: 10.3934/jmd.2021004 +[Abstract](228) +[HTML](100) +[PDF](320.14KB)
Abstract:

Let \begin{document}$G$\end{document} be the group of orientation-preserving isometries of a rank-one symmetric space \begin{document}$X$\end{document} of non-compact type. We study local rigidity of certain actions of a solvable subgroup \begin{document}$\Gamma \subset G$\end{document} on the boundary of \begin{document}$X$\end{document}, which is diffeomorphic to a sphere. When \begin{document}$X$\end{document} is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.

2021, 17: 145-182 doi: 10.3934/jmd.2021005 +[Abstract](248) +[HTML](96) +[PDF](330.08KB)
Abstract:

Given a countable amenable group \begin{document}$G$\end{document} and \begin{document}$\lambda \in (0,1)$\end{document}, we give an elementary construction of a type-Ⅲ\begin{document}$_{\lambda}$\end{document} Bernoulli group action. In the case where \begin{document}$G$\end{document} is the integers, we show that our nonsingular Bernoulli shifts have independent and identically distributed factors.

2021, 17: 183-211 doi: 10.3934/jmd.2021006 +[Abstract](80) +[HTML](35) +[PDF](264.83KB)
Abstract:

We generalize the notion of cusp excursion of geodesic rays by introducing for any \begin{document}$k\geq 1$\end{document} the \begin{document}$k^\text{th}$\end{document} excursion in the cusps of a hyperbolic \begin{document}$N$\end{document}-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk. On the other hand, for \begin{document}$k = N-1$\end{document}, the \begin{document}$k^\text{th}$\end{document} excursion is superlinear for geodesics that are generic with respect to the Lebesgue measure. We use this to show that the hitting measure and the Lebesgue measure on the boundary of hyperbolic space \begin{document}$\mathbb{H}^N$\end{document} for any \begin{document}$N \geq 2$\end{document} are mutually singular.

2021, 17: 213-265 doi: 10.3934/jmd.2021007 +[Abstract](75) +[HTML](32) +[PDF](477.35KB)
Abstract:

We construct a rigid, rank 1, prime transformation that is not quasi-simple and whose self-joinings form a Poulsen simplex. This seems to be the first example of a prime system whose self-joinings form a Poulsen simplex.

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