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Journal of Modern Dynamics

 2022 , Volume 18

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A new dynamical proof of the Shmerkin–Wu theorem
Tim Austin
2022, 18: 1-11 doi: 10.3934/jmd.2022001 +[Abstract](759) +[HTML](116) +[PDF](162.97KB)

Let \begin{document}$ a < b $\end{document} be multiplicatively independent integers, both at least \begin{document}$ 2 $\end{document}. Let \begin{document}$ A,B $\end{document} be closed subsets of \begin{document}$ [0,1] $\end{document} that are forward invariant under multiplication by \begin{document}$ a $\end{document}, \begin{document}$ b $\end{document} respectively, and let \begin{document}$ C : = A\times B $\end{document}. An old conjecture of Furstenberg asserted that any planar line \begin{document}$ L $\end{document} not parallel to either axis must intersect \begin{document}$ C $\end{document} in Hausdorff dimension at most \begin{document}$ \max\{\dim C,1\} - 1 $\end{document}. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.

New time-changes of unipotent flows on quotients of Lorentz groups
Siyuan Tang
2022, 18: 13-67 doi: 10.3934/jmd.2022002 +[Abstract](553) +[HTML](105) +[PDF](444.8KB)

We study the cocompact lattices \begin{document}$ \Gamma\subset SO(n, 1) $\end{document} so that the Laplace–Beltrami operator \begin{document}$ \Delta $\end{document} on \begin{document}$ SO(n)\backslash SO(n, 1)/\Gamma $\end{document} has eigenvalues in \begin{document}$ (0, \frac{1}{4}) $\end{document}, and then show that there exist time-changes of unipotent flows on \begin{document}$ SO(n, 1)/\Gamma $\end{document} that are not measurably conjugate to the unperturbed ones. A main ingredient of the proof is a stronger version of the branching of the complementary series. Combining it with a refinement of the works of Ratner and Flaminio–Forni is adequate for our purpose.

Higher bifurcations for polynomial skew products
Matthieu Astorg and Fabrizio Bianchi
2022, 18: 69-99 doi: 10.3934/jmd.2022003 +[Abstract](295) +[HTML](82) +[PDF](337.57KB)

We continue our investigation of the parameter space of families of polynomial skew products. Assuming that the base polynomial has a Julia set not totally disconnected and is neither a Chebyshev nor a power map, we prove that, near any bifurcation parameter, one can find parameters where \begin{document}$ k $\end{document} critical points bifurcate independently, with \begin{document}$ k $\end{document} up to the dimension of the parameter space. This is a striking difference with respect to the one-dimensional case. The proof is based on a variant of the inclination lemma, applied to the postcritical set at a Misiurewicz parameter. By means of an analytical criterion for the non-vanishing of the self-intersections of the bifurcation current, we deduce the equality of the supports of the bifurcation current and the bifurcation measure for such families. Combined with results by Dujardin and Taflin, this also implies that the support of the bifurcation measure in these families has non-empty interior. As part of our proof we construct, in these families, subfamilies of codimension 1 where the bifurcation locus has non empty interior. This provides a new independent proof of the existence of holomorphic families of arbitrarily large dimension whose bifurcation locus has non empty interior. Finally, it shows that the Hausdorff dimension of the support of the bifurcation measure is maximal at any point of its support.

The 2020 Michael Brin Prize in Dynamical Systems
The Editors
2022, 18: 101-102 doi: 10.3934/jmd.2022004 +[Abstract](444) +[HTML](140) +[PDF](4787.88KB)
Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai
Mariusz Lemańczyk
2022, 18: 103-130 doi: 10.3934/jmd.2022005 +[Abstract](274) +[HTML](99) +[PDF](290.72KB)

We present and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.

Some arithmetical aspects of renormalization in Teichmüller dynamics: On the occasion of Corinna Ulcigrai winning the Brin Prize
Stefano Marmi
2022, 18: 131-147 doi: 10.3934/jmd.2022006 +[Abstract](249) +[HTML](68) +[PDF](477.38KB)

We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmüller dynamics.

Hodge and Teichmüller
Jeremy Kahn and Alex Wright
2022, 18: 149-160 doi: 10.3934/jmd.2022007 +[Abstract](219) +[HTML](47) +[PDF](181.27KB)

We consider the derivative \begin{document}$ D\pi $\end{document} of the projection \begin{document}$ \pi $\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$ \eta $\end{document} determines a relative cohomology class \begin{document}$ [\eta]_\Sigma $\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$ D\pi([\eta]_\Sigma) $\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.

Eigenvalue gaps for hyperbolic groups and semigroups
Fanny Kassel and Rafael Potrie
2022, 18: 161-208 doi: 10.3934/jmd.2022008 +[Abstract](125) +[HTML](25) +[PDF](469.99KB)

Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the \begin{document}$ i^\text{th} $\end{document} and \begin{document}$ (i+1)^\text{th} $\end{document} Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index \begin{document}$ i $\end{document}. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence
Dmitry Dolgopyat, Bassam Fayad and Sixu Liu
2022, 18: 209-289 doi: 10.3934/jmd.2022009 +[Abstract](68) +[HTML](24) +[PDF](604.99KB)

A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.

2021 Impact Factor: 0.641
5 Year Impact Factor: 0.894
2021 CiteScore: 1.1


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