The multiple Birkhoff recurrence theorem states that for any $dinmathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,ldots, x)$ is recurrent under $tau_d=:Ttimes T^2times ldots times T^d$. It is natural to ask if there alwa
ys is a multiply minimal point, i.e. a point $x$ such that $(x,x,ldots,x)$ is $tau_d$-minimal. A negative answer is presented in this paper via studying the horocycle flows. However, it is shown that for any minimal system $(X,T)$ and any non-empty open set $U$, there is $xin U$ such that ${nin{mathbb Z}: T^nxin U, ldots, T^{dn}xin U}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, tau_d)$ is minimal.
We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$,
the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{{mathcal{O}}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. Using results derived from the above fact, we are able to answer several open questions: (1) if $(X,T^k)$ is minimal for some $kge 2$, then for any $din {mathbb N}$ and any $0le j<k$ there is a sequence ${n_i}$ of $mathbb Z$ with $n_iequiv j (text{mod} k)$ such that $T^{n_i}xrightarrow x, T^{2n_i}xrightarrow x, ldots, T^{dn_i}xrightarrow x$ for $x$ in a dense $G_delta$ subset of $X$; (2) if $(X,T)$ is totally minimal, then ${T^{n^2}x:nin {mathbb Z}}$ is dense in $X$ for $x$ in a dense $G_delta$ subset of $X$; (3) for any $dinmathbb N$ and any minimal system, which is an open extension of its maximal distal factor, ${bf RP}^{[d]}={bf AP}^{[d]}$, where the latter is the regionally proximal relation of order $d$ along arithmetic progressions.
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 orb
it 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 $d$.
In this paper it is proved that if a minimal system has the property that its sequence entropy is uniformly bounded for all sequences, then it has only finitely many ergodic measures and is an almost finite to one extension of its maximal equicontinu
ous factor. This result is obtained as an application of a general criteria which states that if a minimal system is an almost finite to one extension of its maximal equicontinuous factor and has no infinite independent sets of length $k$ for some $kge 2$, then it has only finitely many ergodic measures.
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 ergo
dic 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.
In this paper, we show that for any sequence ${bf a}=(a_n)_{nin Z}in {1,ldots,k}^mathbb{Z}$ and any $epsilon>0$, there exists a Toeplitz sequence ${bf b}=(b_n)_{nin Z}in {1,ldots,k}^mathbb{Z}$ such that the entropy $h({bf b})leq 2 h({bf a})$ and $lim
_{Ntoinfty}frac{1}{2N+1}sum_{n=-N}^N|a_n-b_n|<epsilon$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the M{o}bius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.
In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for
a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.
In this paper we give an answer to Furstenbergs problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subse
t $D$ of $X$ such that for any minimal system $(Y,S)$, any point $yin Y$ and any open neighbourhood $V$ of $y$, and for any nonempty open subset $Usubset X$, there is $xin Dcap U$ satisfying that ${nin{ mathbb Z}_+: T^nxin U, S^nyin V}$ is syndetic. Some characterization for the general case is also described. As applications we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^n,T^{(n)})$ and $(X, T^n)$ for any $nin { mathbb N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_K)$ is disjoint from all minimal systems.
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether
mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages begin{equation*} frac 1 Nsum_{n=0}^{N-1}f_1(T^nx)cdots f_d(T^{dn}x), quad Nto infty, end{equation*} almost surely converge.
It is shown that any non-PI minimal system is Li-Yorke sensitive. Consequently, any minimal system with nontrivial weakly mixing factor (such a system is non-PI) is Li-Yorke sensitive, which answers affirmatively an open question by Akin and Kolyada.
