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Register a new userGeometric and Measure-Theoretic Shrinking Targets in Dynamical Systems
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We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.
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We investigate Weierstrass functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H={loggamma}/{log frac12}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough $gamma$ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.
We establish the Geometric Dynamical Northcott Property for polarized endomorphisms of a projective normal variety over a function field $mathbf{K}$ of characteristic zero. This extends previous results of Benedetto, Baker and DeMarco in dimension $1$, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous one. We first show that when $mathbf{K}$ is the field of rational functions of a smooth complex projective variety, the canonical height of a subvariety is the mass of the appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument. Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the Geometric Bogomolov Conjecture, recently proved by Cantat, Gao, Habegger and Xie. We establish several cases of this conjecture notably non-isotrivial polynomial skew-product with an isotrivial first coordinate.
In this paper we prove the existence of a simultaneous local normalization for couples $(X,mathcal{G})$, where $X$ is a vector field which vanishes at a point and $mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $mathcal{G}$ is singular but $X$ does not vanish at the origin.
We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: $cdot$ (Existence of density-1 UI and OI set) Let $(X,mathcal{B},mu,T)$ be an invertible probability measure preserving weakly mixing system. Then for any $dinmathbb{N}$, any non-constant integer-valued polynomials $p_{1},p_{2},dots,p_{d}$ such that $p_{i}-p_{j}$ are also non-constant for all $i eq j$, (i) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)<mu(A)^{d+1}}$$ is of density 1. (ii) there is $Ainmathcal{B}$ such that the set $${ninmathbb{N}colonmu(Acap T^{p_{1}(n)}Acapdotscap T^{p_{d}(n)}A)>mu(A)^{d+1}}$$ is of density 1. $cdot$ (Existence of Ces`aro OI set) Let $(X,mathcal{B},mu,T)$ be a free, invertible, ergodic probability measure preserving system and $Minmathbb{N}$. %Suppose that $X$ contains an ergodic component which is aperiodic. Then there is $Ainmathcal{B}$ such that $$frac{1}{N}sum_{n=M}^{N+M-1}mu(Acap T^{n}A)>mu(A)^{2}$$ for all $Ninmathbb{N}$. $cdot$ (Nonexistence of Ces`aro UI set) Let $(X,mathcal{B},mu,T)$ be an invertible probability measure preserving system. For any measurable set $A$ satisfying $mu(A) in (0,1)$, there exist infinitely many $N in mathbb{N}$ such that $$frac{1}{N} sum_{n=0}^{N-1} mu ( A cap T^{n}A) > mu(A)^2.$$
We investigate Takagi-type functions with roughness parameter $gamma$ that are Holder continuous with coefficient $H=frac{loggamma}{log eh}.$ Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We show that the SBR measure is absolutely continuous for large enough $gamma$. Dually, where duality is related to time reversal, we prove that for large enough $gamma$ a version of the Takagi-type curve centered around fibers of the associated stable manifold possesses a square integrable local time.
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