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Journal of Modern Dynamics

August 2017 , Volume 11

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Logarithm laws for unipotent flows, Ⅱ
Jayadev S. Athreya and Gregory A. Margulis
2017, 11: 1-16 doi: 10.3934/jmd.2017001 +[Abstract](2935) +[HTML](63) +[PDF](186.6KB)
Abstract:

We prove analogs of the logarithm laws of Sullivan and KleinbockMargulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces G/Γ.

The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve
Matteo Costantini and André Kappes
2017, 11: 17-41 doi: 10.3934/jmd.2017002 +[Abstract](1507) +[HTML](60) +[PDF](255.8KB)
Abstract:

We compute the algebraic equation of the universal family over the Kenyon-Smillie (2, 3, 4)-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

Positive metric entropy in nondegenerate nearly integrable systems
Dong Chen
2017, 11: 43-56 doi: 10.3934/jmd.2017003 +[Abstract](1537) +[HTML](25) +[PDF](196.7KB)
Abstract:

The celebrated KAM theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^\infty$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori.

Distribution of postcritically finite polynomials Ⅱ: Speed of convergence
Thomas Gauthier and Gabriel Vigny
2017, 11: 57-98 doi: 10.3934/jmd.2017004 +[Abstract](2637) +[HTML](43) +[PDF](388.2KB)
Abstract:

In the moduli space of degree $d$ polynomials, we prove the equidistribution of postcritically finite polynomials toward the bifurcation measure. More precisely, using complex analytic arguments and pluripotential theory, we prove the exponential speed of convergence for $\mathscr{C}^2$-observables. This improves results obtained with arithmetic methods by Favre and Rivera-Letellier in the unicritical family and Favre and the first author in the space of degree $d$ polynomials.

We deduce from that the equidistribution of hyperbolic parameters with $(d-1)$ distinct attracting cycles of given multipliers toward the bifurcation measure with exponential speed for $\mathscr{C}^1$-observables. As an application, we prove the equidistribution (up to an explicit extraction) of parameters with $(d-1)$ distinct cycles with prescribed multiplier toward the bifurcation measure for any $(d-1)$ multipliers outside a pluripolar set.

On spectra of Koopman, groupoid and quasi-regular representations
Artem Dudko and Rostislav Grigorchuk
2017, 11: 99-123 doi: 10.3934/jmd.2017005 +[Abstract](1441) +[HTML](32) +[PDF](282.7KB)
Abstract:

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups
Nikolaos Karaliolios
2017, 11: 125-142 doi: 10.3934/jmd.2017006 +[Abstract](1443) +[HTML](28) +[PDF](223.8KB)
Abstract:

We study close-to-constants quasiperiodic cocycles in \begin{document} $\mathbb{T} ^{d} \times G$ \end{document}, where \begin{document} $d \in \mathbb{N} ^{*} $ \end{document} and \begin{document} $G$ \end{document} is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is \begin{document} $C^{\infty}$ \end{document}-conjugate to it, and the KAM scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.

Approximation of points in the plane by generic lattice orbits
Dubi Kelmer
2017, 11: 143-153 doi: 10.3934/jmd.2017007 +[Abstract](2796) +[HTML](60) +[PDF](183.1KB)
Abstract:

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice \begin{document}$\Gamma < {\rm{S}}{{\rm{L}}_2}\left( {\mathbb{R}} \right)$\end{document} acting linearly on \begin{document}${\mathbb{R}^2}$\end{document}. Our method gives bounds that are uniform for almost all orbits.

On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)
Samuel C. Edwards
2017, 11: 155-188 doi: 10.3934/jmd.2017008 +[Abstract](470) +[HTML](1) +[PDF](326.2KB)
Abstract:

Let \begin{document}$\Gamma$\end{document} be a lattice in \begin{document}$G=\mathrm{SL}(2, \mathbb{C})$\end{document}. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in \begin{document}$\Gamma\backslash G$\end{document}. Our method of proof relies on the theory of unitary representations.

On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$)
Samuel C. Edwards
2017, 11: 155-188 doi: 10.3934/jmd.2013.7.291 +[Abstract](411) +[HTML](31) +[PDF](400.4KB)
Abstract:

Let \begin{document}$\Gamma$\end{document} be a lattice in \begin{document}$G=\mathrm{SL}(2, \mathbb{C})$\end{document}. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in \begin{document}$\Gamma\backslash G$\end{document}. Our method of proof relies on the theory of unitary representations.

Effective equidistribution of circles in the limit sets of Kleinian groups
Wenyu Pan
2017, 11: 189-217 doi: 10.3934/jmd.2017009 +[Abstract](2721) +[HTML](70) +[PDF](386.4KB)
Abstract:

Consider a general circle packing \begin{document}$\mathscr{P}$\end{document} in the complex plane \begin{document}$\mathbb{C}$\end{document} invariant under a Kleinian group \begin{document}$\Gamma$\end{document}. When \begin{document}$\Gamma$\end{document} is convex cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in \begin{document}$\mathscr{P}$\end{document} intersecting any bounded connected regular set in \begin{document}$\mathbb{C}$\end{document}; this provides an effective version of an earlier work of Oh-Shah [12]. In view of the recent result of McMullen-Mohammadi-Oh [6], our effective circle counting theorem applies to the circles contained in the limit set of a convex cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover, consider the circle packing \begin{document}$\mathscr{P}(\mathscr{T})$\end{document} of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than \begin{document}$t$\end{document}, as \begin{document}$t\to0$\end{document}, effectivizing the work of Oh [10].

Minimality of interval exchange transformations with restrictions
Ivan Dynnikov and Alexandra Skripchenko
2017, 11: 219-248 doi: 10.3934/jmd.2017010 +[Abstract](1586) +[HTML](37) +[PDF](685.9KB)
Abstract:

It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this occurs if and only if the linear restrictions contain a Lagrangian subspace of the first homology of the suspension surface. We partially prove it in the `only if' direction and provide a series of examples to support the converse one. We show that the unique ergodicity remains a generic property if the restrictions on the parameters do not contain a Lagrangian subspace (this result is due to Barak Weiss).

Most interval exchanges have no roots
Daniel Bernazzani
2017, 11: 249-262 doi: 10.3934/jmd.2017011 +[Abstract](1449) +[HTML](31) +[PDF](183.8KB)
Abstract:

Let \begin{document}$T$\end{document} be an \begin{document}$m$\end{document}-interval exchange transformation. By the rank of \begin{document}$T$\end{document} we mean the dimension of the \begin{document}$\mathbb{Q}$\end{document}-vector space spanned by the lengths of the exchanged intervals. We prove that if \begin{document}$T$\end{document} is minimal and the rank of \begin{document}$T$\end{document} is greater than \begin{document}$1+\lfloor m/2 \rfloor$\end{document}, then \begin{document}$T$\end{document} cannot be written as a power of another interval exchange. We also demonstrate that this estimate on the rank cannot be improved.

In the case that \begin{document}$T$\end{document} is a minimal 3-interval exchange transformation, we prove a stronger result: \begin{document}$T$\end{document} cannot be written as a power of another interval exchange if and only if \begin{document}$T$\end{document} satisfies Keane's infinite distinct orbit condition. In the course of proving this result, we give a classification (up to conjugacy) of those minimal interval exchange transformations whose discontinuities all belong to a single orbit.

Exponential mixing and smooth classification of commuting expanding maps
Ralf Spatzier and Lei Yang
2017, 11: 263-312 doi: 10.3934/jmd.2017012 +[Abstract](1494) +[HTML](38) +[PDF](456.5KB)
Abstract:

We show that genuinely higher rank expanding actions of abelian semigroups on compact manifolds are \begin{document}$C.{\infty}$\end{document}-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to Hölder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semigroup actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.

Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories
Morimichi Kawasaki and Ryuma Orita
2017, 11: 313-339 doi: 10.3934/jmd.2017013 +[Abstract](2755) +[HTML](47) +[PDF](425.1KB)
Abstract:

The first author [9] introduced a relative symplectic capacity \begin{document}$C$\end{document} for a symplectic manifold \begin{document}$(N,\omega_N)$\end{document} and its subset \begin{document}$X$\end{document} which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of \begin{document}$N$\end{document} with the annulus \begin{document}$A_R=(-R,R)\times\mathbb{R}/\mathbb{Z}$\end{document}. In the present paper, we give an exact computation of the capacity \begin{document}$C$\end{document} of the \begin{document}$2n$\end{document}-torus \begin{document}$\mathbb{T}^{2n}$\end{document} relative to a Lagrangian submanifold \begin{document}$\mathbb{T}^n$\end{document} which implies the existence of non-contractible Hamiltonian periodic trajectories on \begin{document}$A_R\times\mathbb{T}^{2n}$\end{document}. Moreover, we give a lower bound on the number of such trajectories.

Normal forms for non-uniform contractions
Boris Kalinin and Victoria Sadovskaya
2017, 11: 341-368 doi: 10.3934/jmd.2017014 +[Abstract](1484) +[HTML](27) +[PDF](287.0KB)
Abstract:

Let \begin{document}$f$\end{document} be a measure-preserving transformation of a Lebesgue space \begin{document}$(X,\mu)$\end{document} and let \begin{document}${\mathscr{F}}$\end{document} be its extension to a bundle \begin{document}$\mathscr{E} = X \times {\mathbb{R}}^m$\end{document} by smooth fiber maps \begin{document}${\mathscr{F}}_x : {\mathscr{E}}_x \to {\mathscr{E}}_{fx}$\end{document} so that the derivative of \begin{document}${\mathscr{F}}$\end{document} at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes \begin{document}${\mathscr{H}}_x$\end{document} on \begin{document}${\mathscr{E}}_x$\end{document} for \begin{document}$\mu$\end{document}-a.e. \begin{document}$x$\end{document} so that the maps \begin{document}${\mathscr{P}}_x ={\mathscr{H}}_{fx} \circ {\mathscr{F}}_x \circ {\mathscr{H}}_x^{-1}$\end{document} are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such \begin{document}${\mathscr{H}}_x$\end{document} and \begin{document}${\mathscr{P}}_x$\end{document} are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change \begin{document}$\mathscr{H}$\end{document} also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism \begin{document}$f$\end{document} with a non-uniformly contracting invariant foliation \begin{document}$W$\end{document}. We construct a measurable system of smooth coordinate changes \begin{document}${\mathscr{H}}_x: W_x \to T_xW$\end{document} such that the maps \begin{document}${\mathscr{H}}_{fx} \circ f \circ {\mathscr{H}}_x^{-1}$\end{document} are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.

Escape of mass in homogeneous dynamics in positive characteristic
Alexander Kemarsky, Frédéric Paulin and Uri Shapira
2017, 11: 369-407 doi: 10.3934/jmd.2017015 +[Abstract](1538) +[HTML](45) +[PDF](1611.2KB)
Abstract:

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of \begin{document}$2$\end{document}-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue: For a large set of rays, the measures fail to converge to the uniform probability measure on the space of \begin{document}$2$\end{document}-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of \begin{document}$2$\end{document}-lattices.

A disconnected deformation space of rational maps
Eriko Hironaka and Sarah Koch
2017, 11: 409-423 doi: 10.3934/jmd.2017016 +[Abstract](1285) +[HTML](37) +[PDF](358.22KB)
Abstract:

The deformation space of a branched cover \begin{document}$f:(S^2,A)\to (S^2,B)$\end{document} is a complex submanifold of a certain Teichmüller space, which consists of classes of marked rational maps \begin{document}$F:(\mathbb{P}^1,A')\to (\mathbb{P}^1,B')$\end{document} that are combinatorially equivalent to \begin{document}$f$\end{document}. In the case \begin{document}$A=B$\end{document}, under a mild assumption on \begin{document}$f$\end{document}, William Thurston gave a topological criterion for which the deformation space of \begin{document}$f:(S^2,A)\to (S^2,B)$\end{document} is nonempty, and he proved that it is always connected. We show that if \begin{document}$A\subsetneq B$\end{document}, then the deformation space need not be connected. We exhibit a family of quadratic rational maps for which the associated deformation spaces are disconnected; in fact, each has infinitely many components.

An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms
Bassam Fayad and Zhiyuan Zhang
2017, 11: 425-445 doi: 10.3934/jmd.2017017 +[Abstract](1687) +[HTML](65) +[PDF](264.84KB)
Abstract:

For area preserving C2 surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's (n, δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.

Logarithm laws for unipotent flows on hyperbolic manifolds
Shucheng Yu
2017, 11: 447-476 doi: 10.3934/jmd.2017018 +[Abstract](1318) +[HTML](33) +[PDF](312.47KB)
Abstract:

We prove logarithm laws for unipotent flows on non-compact finite-volume hyperbolic manifolds. Our method depends on the estimate of norms of certain incomplete Eisenstein series.

The gap distribution of directions in some Schottky groups
Xin Zhang
2017, 11: 477-499 doi: 10.3934/jmd.2017019 +[Abstract](1306) +[HTML](37) +[PDF](632.75KB)
Abstract:

We prove the existence and some properties of the limiting gap distribution for the directions of some Schottky group orbits in the Poincaré disk. A key feature is that the fundamental domains for these groups have infinite area.

Asymptotic distribution of values of isotropic here quadratic forms at S-integral points
Jiyoung Han, Seonhee Lim and Keivan Mallahi-Karai
2017, 11: 501-550 doi: 10.3934/jmd.2017020 +[Abstract](891) +[HTML](37) +[PDF](464.61KB)
Abstract:

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places \begin{document} $S$ \end{document} over \begin{document} ${\mathbb{Q}}$ \end{document} containing the Archimedean place and excluding the prime \begin{document} $2$ \end{document}, an irrational isotropic form \begin{document} ${\mathbf q}$ \end{document} of rank \begin{document} $n\geq 4$ \end{document} on \begin{document} ${\mathbb{Q}}_S$ \end{document}, a product of \begin{document} $p$ \end{document}-adic intervals \begin{document} $\mathsf{I}_p$ \end{document}, and a product \begin{document} $\Omega$ \end{document} of star-shaped sets. We show that unless \begin{document} $n=4$ \end{document} and \begin{document} ${\mathbf q}$ \end{document} is split in at least one place, the number of \begin{document} $S$ \end{document}-integral vectors \begin{document} $\mathbf v \in {\mathsf{T}} \Omega$ \end{document} satisfying simultaneously \begin{document} ${\mathbf q}(\mathbf v) \in I_p$ \end{document} for \begin{document} $p \in S$ \end{document} is asymptotically given by

as \begin{document} ${\mathsf{T}}$ \end{document} goes to infinity, where \begin{document} $|\,\mathsf{I}\,|$ \end{document} is the product of Haar measures of the \begin{document} $p$ \end{document}-adic intervals \begin{document} $I_p$ \end{document}. The proof uses dynamics of unipotent flows on \begin{document} $S$ \end{document}-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an \begin{document} $S$ \end{document}-arithmetic variant of the \begin{document} $ \alpha$ \end{document}-function introduced in [10], and an \begin{document} $S$ \end{document}-arithemtic version of a theorem of Dani-Margulis [7].

Fourier coefficients of $\times p$-invariant measures
Huichi Huang
2017, 11: 551-562 doi: 10.3934/jmd.2017021 +[Abstract](1352) +[HTML](42) +[PDF](163.67KB)
Abstract:

We consider densities \begin{document}$D_\Sigma(A)$\end{document}, \begin{document}$\overline{D}_\Sigma(A)$\end{document} and \begin{document}$\underline{D}_\Sigma(A)$\end{document} for a subset \begin{document}$A$\end{document} of \begin{document}$\mathbb{N}$\end{document} with respect to a sequence \begin{document}$\Sigma$\end{document} of finite subsets of \begin{document}$\mathbb{N}$\end{document} and study Fourier coefficients of ergodic, weakly mixing and strongly mixing \begin{document}$\times p$\end{document}-invariant measures on the unit circle \begin{document}$\mathbb{T}$\end{document}. Combining these, we prove the following measure rigidity results: on \begin{document}$\mathbb{T}$\end{document}, the Lebesgue measure is the only non-atomic \begin{document}$\times p$\end{document}-invariant measure satisfying one of the following: (1) \begin{document}$\mu$\end{document} is ergodic and there exist a Følner sequence \begin{document}$\Sigma$\end{document} in \begin{document}$\mathbb{N}$\end{document} and a nonzero integer \begin{document}$l$\end{document} such that \begin{document}$\mu$\end{document} is \begin{document}$\times (p^j+l)$\end{document}-invariant for all \begin{document}$j$\end{document} in a subset \begin{document}$A$\end{document} of \begin{document}$\mathbb{N}$\end{document} with \begin{document}$D_\Sigma(A)=1$\end{document}; (2) \begin{document}$\mu$\end{document} is weakly mixing and there exist a Følner sequence \begin{document}$\Sigma$\end{document} in \begin{document}$\mathbb{N}$\end{document} and a nonzero integer \begin{document}$l$\end{document} such that \begin{document}$\mu$\end{document} is \begin{document}$\times (p^j+l)$\end{document}-invariant for all \begin{document}$j$\end{document} in a subset \begin{document}$A$\end{document} of \begin{document}$\mathbb{N}$\end{document} with \begin{document}$\overline{D}_\Sigma(A)>0$\end{document}; (3) \begin{document}$\mu$\end{document} is strongly mixing and there exists a nonzero integer \begin{document}$l$\end{document} such that \begin{document}$\mu$\end{document} is \begin{document}$\times (p^j+l)$\end{document}-invariant for infinitely many \begin{document}$j$\end{document}. Moreover, a \begin{document}$\times p$\end{document}-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure.

As an application we prove that for every increasing function \begin{document}$\tau$\end{document} defined on positive integers with \begin{document}$\lim_{n\to\infty}\tau(n)=\infty$\end{document}, there exists a multiplicative semigroup \begin{document}$S_\tau$\end{document} of \begin{document}$\mathbb{Z}^+$\end{document} containing \begin{document}$p$\end{document} such that \begin{document}$|S_\tau\cap[1,n]|\leq (\log_p n)^{\tau(n)}$\end{document} and the Lebesgue measure is the only non-atomic ergodic \begin{document}$\!\times \!p$\end{document}-invariant measure which is \begin{document}$\times q$\end{document}-invariant for all \begin{document}$q$\end{document} in \begin{document}$S_\tau$\end{document}.

Logarithmic laws and unique ergodicity
Jon Chaika and Rodrigo Treviño
2017, 11: 563-588 doi: 10.3934/jmd.2017022 +[Abstract](2530) +[HTML](55) +[PDF](277.58KB)
Abstract:

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

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