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Register a new userPro-nilfactors of the space of arithmetic progressions in topological dynamical systems
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For a topological dynamical system $(X, T)$, $linmathbb{N}$ and $xin X$, let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point $(x,x,ldots,x)$ ($l $ times) under the actions $mathcal{G}_{l}$ and $tau_l $ respectively, where $mathcal{G}_{l}$ is generated by $Ttimes Ttimes ldots times T$ ($l $ times) and $tau_l=Ttimes T^2times ldots times T^l$. In this paper, we show that for a minimal system $(X,T)$ and $lin mathbb{N}$, the maximal $d$-step pro-nilfactor of $(N_l(X),mathcal{G}_{l})$ is $(N_l(X_d),mathcal{G}_{l})$, where $pi_d:Xto X/mathbf{RP}^{[d]}=X_d,din mathbb{N}$ is the factor map and $mathbf{RP}^{[d]}$ is the regionally proximal relation of order $d$. Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the pro-nilfactors of $(L_x^l(X),tau_l)$ for almost every $x$ w.r.t. the Haar measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a countable set $Omegasubset Y$ such that for $yin Ybackslash Omega$ the maximal equicontinuous factor of $(L_y^2(Y),tau_2)$ is not $(L_{pi_1(y)}^2(Y_{1}),tau_2)$.
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Let $pi: (X,T)rightarrow (Y,T)$ be a factor map of topological dynamics and $din {mathbb {N}}$. $(Y,T)$ is said to be a $d$-step topological characteristic factor if there exists a dense $G_delta$ set $X_0$ of $X$ such that for each $xin X_0$ the orbit closure $overline{mathcal O}((x, ldots,x), Ttimes T^2times ldots times T^d)$ is $pitimes ldots times pi$ ($d$ times) saturated. In 1994 Eli Glasner studied the topological characteristic factor for minimal systems. For example, it is shown that for a distal minimal system, its largest distal factor of order $d-1$ is its $d$-step topological characteristic factor. In this paper, we generalize Glasners work to the product system of finitely many minimal systems and give its relative version. To prove these results, we need to deal with $(X,T^m)$ for $min {mathbb {N}}$. We will study the structure theorem of $(X,T^m)$. We show that though for a minimal system $(X,T)$ and $min {mathbb {N}}$, $(X,T^m)$ may not be minimal, but we still can have PI-tower for $(X,T^m)$ and in fact it looks the same as the PI tower of $(X,T)$. We give some applications of the results developed. For example, we show that if a minimal system has no nontrivial independent pair along arithmetic progressions of order $d$, then up to a canonically defined proximal extension, it is PI of order $d$; if a minimal system $(X,T)$ has a nontrivial $d$-step topological characteristic factor, then there exist ``many $Delta$-transitive sets of order $d$.
The regionally proximal relation of order $d$ along arithmetic progressions, namely ${bf AP}^{[d]}$ for $din N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${bf AP}^{[d]}=Delta$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. It follows that for a minimal $infty$-pro-nilsystem, ${bf AP}^{[d]}={bf RP}^{[d]}$ for each $din {mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andre-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
Let $mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,{f_n}_{n=1}^{+infty})$ vanishes, then so does that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$; moreover, once the topological entropy of $(X,{f_n}_{n=1}^{+infty})$ is positive, that of its induced system $(mathcal{M}(X),{f_n}_{n=1}^{+infty})$ jumps to infinity. In contrast to Bowens inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.
We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnaks conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe and for all toral affine dynamical systems of positive entropy, some of which dont have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.
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