Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{\Omega(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicities. We study the set $Acc^\Omega(x)$ of weak-$^*$ limits of the sequence $\frac{1}{N}\sum_{n\leq N}\delta_{T^{\Omega(n)}x}$ in $\sigma$-compact dynamical systems $ (X,T)$, demonstrating that if $x \in X$ is quasi-generic for an ergodic...
Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $\Omega (|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergels...
Let $(X,T)$ be a topological dynamical system, and let $\dim(X,T)$ denote Meyerovitch's dynamical dimension. We prove that, for every integer $n\geq1$, if $\dim(X,T)<n/2$, then the set of maps $f\in C(X,[0,1]^n)$ for which the orbit map $$ I_f: X\longrightarrow([0,1]^n)^{\mathbb{Z}}, \qquad I_f(x)=\bigl(f(T^k x)\bigr)_...
We study the skew product systems $T: \mathbb{S}^1\times\mathbb{R}^d \to \mathbb{S}^1 \times\mathbb{R}^d$, \begin{equation*} T(x,y)=(\ell x, A y+\phi(x)), \end{equation*} where $\ell\geq 2,$ $A \in GL_d(\mathbb{R})$ with $\rho(A)<1$, and $\phi \in C^r(\mathbb{S}^1, \mathbb{R}^d)$. We allow the fibers to have any dimens...
D. Galli, E. Passaglia, Andrea Ulliana· 1 citation
We study the diagonal Green function $\widetilde{u}(x)=[L_N^{-1}]_{x,x}$ of the operator $L_N=I-\alpha P$ on the finite torus $(\mathbb{Z}/N\mathbb{Z})^2$, where $P$ is the transfer operator of the discrete cat map $T_N(x)=Ax \bmod N$. We prove the exact formula $\widetilde{u}(x)=(1-\alpha^{k_x})^{-1}$, where $k_x$ is...
When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ wit...
Xavier Buff, Valentin Huguin, L. Vivas· 0 citations
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