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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](941) +[HTML](412) +[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](800) +[HTML](369) +[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](696) +[HTML](356) +[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.

Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces
Mao Okada
2021, 17: 111-143 doi: 10.3934/jmd.2021004 +[Abstract](678) +[HTML](320) +[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.

The orbital equivalence of Bernoulli actions and their Sinai factors
Zemer Kosloff and Terry Soo
2021, 17: 145-182 doi: 10.3934/jmd.2021005 +[Abstract](790) +[HTML](308) +[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.

Cusp excursion in hyperbolic manifolds and singularity of harmonic measure
Anja Randecker and Giulio Tiozzo
2021, 17: 183-211 doi: 10.3934/jmd.2021006 +[Abstract](516) +[HTML](269) +[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.

A prime system with many self-joinings
Jon Chaika and Bryna Kra
2021, 17: 213-265 doi: 10.3934/jmd.2021007 +[Abstract](537) +[HTML](249) +[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.

Tri-Coble surfaces and their automorphisms
John Lesieutre
2021, 17: 267-284 doi: 10.3934/jmd.2021008 +[Abstract](516) +[HTML](228) +[PDF](503.32KB)
Abstract:

We construct some positive entropy automorphisms of rational surfaces with no periodic curves. The surfaces in question, which we term tri-Coble surfaces, are blow-ups of \begin{document}$ \mathbb P^2$\end{document} at 12 points which have contractions down to three different Coble surfaces. The automorphisms arise as compositions of lifts of Bertini involutions from certain degree \begin{document}$1$\end{document} weak del Pezzo surfaces.

Direct products, overlapping actions, and critical regularity
Sang-hyun Kim, Thomas Koberda and Cristóbal Rivas
2021, 17: 285-304 doi: 10.3934/jmd.2021009 +[Abstract](920) +[HTML](102) +[PDF](259.54KB)
Abstract:

We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if \begin{document}$ H $\end{document} and \begin{document}$ K $\end{document} are two non-solvable groups then a faithful \begin{document}$ C^{1,\tau} $\end{document} action of \begin{document}$ H\times K $\end{document} on a compact interval \begin{document}$ I $\end{document} is not overlapping for all \begin{document}$ \tau>0 $\end{document}, which by definition means that there must be non-trivial \begin{document}$ h\in H $\end{document} and \begin{document}$ k\in K $\end{document} with disjoint support. As a corollary we prove that the right-angled Artin group \begin{document}$ (F_2\times F_2)*\mathbb{Z} $\end{document} has critical regularity one, which is to say that it admits a faithful \begin{document}$ C^1 $\end{document} action on \begin{document}$ I $\end{document}, but no faithful \begin{document}$ C^{1,\tau} $\end{document} action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group \begin{document}$ F $\end{document} does not admit a faithful \begin{document}$ C^1 $\end{document} overlapping action on \begin{document}$ I $\end{document}, so that \begin{document}$ F*\mathbb{Z} $\end{document} is a new example of a locally indicable group admitting no faithful \begin{document}$ C^1 $\end{document} action on \begin{document}$ I $\end{document}.

Non-autonomous curves on surfaces
Michael Khanevsky
2021, 17: 305-317 doi: 10.3934/jmd.2021010 +[Abstract](492) +[HTML](310) +[PDF](179.29KB)
Abstract:

Consider a symplectic surface \begin{document}$ \Sigma $\end{document} with two properly embedded Hamiltonian isotopic curves \begin{document}$ L $\end{document} and \begin{document}$ L' $\end{document}. Suppose \begin{document}$ g \in Ham(\Sigma) $\end{document} is a Hamiltonian diffeomorphism which sends \begin{document}$ L $\end{document} to \begin{document}$ L' $\end{document}. Which dynamical properties of \begin{document}$ g $\end{document} can be detected by the pair \begin{document}$ (L, L') $\end{document}? We present two scenarios where one can deduce that \begin{document}$ g $\end{document} is "chaotic:" non-autonomous or even of positive entropy.

On the relation between action and linking
David Bechara Senior, Umberto L. Hryniewicz and Pedro A. S. Salomão
2021, 17: 319-336 doi: 10.3934/jmd.2021011 +[Abstract](442) +[HTML](118) +[PDF](216.82KB)
Abstract:

We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.

Horospherically invariant measures and finitely generated Kleinian groups
Or Landesberg
2021, 17: 337-352 doi: 10.3934/jmd.2021012 +[Abstract](320) +[HTML](106) +[PDF](254.03KB)
Abstract:

Let \begin{document}$ \Gamma < {\rm{PSL}}_2( \mathbb{C}) $\end{document} be a Zariski dense finitely generated Kleinian group. We show all Radon measures on \begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document} which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [18] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [2] and Calegari-Gabai [10].

Computing the Rabinowitz Floer homology of tentacular hyperboloids
Alexander Fauck, Will J. Merry and Jagna Wiśniewska
2021, 17: 353-399 doi: 10.3934/jmd.2021013 +[Abstract](321) +[HTML](56) +[PDF](420.7KB)
Abstract:

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids \begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}. Using an embedding of a compact sphere \begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document} into the hypersurface \begin{document}$ \Sigma $\end{document}, we construct a chain map from the Floer complex of \begin{document}$ \Sigma $\end{document} to the Floer complex of \begin{document}$ \Sigma_0 $\end{document}. In contrast to the compact case, the Rabinowitz Floer homology groups of \begin{document}$ \Sigma $\end{document} are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds
Dubi Kelmer and Hee Oh
2021, 17: 401-434 doi: 10.3934/jmd.2021014 +[Abstract](80) +[HTML](24) +[PDF](329.62KB)
Abstract:

Let \begin{document}$ \mathscr{M} $\end{document} be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.

2020 Impact Factor: 0.848
5 Year Impact Factor: 0.815
2020 CiteScore: 0.9

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