In this paper, a polynomial version of the Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct product of an infinite-step pro-nilsystem and a Bernoulli system. As applications, some new convergence theorems are obtained. Particularly, it is proved that if T and S are ergodic measure-preserving transformations on a probability space (X, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\cal{X}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">X</mi> </mrow> </math></EquationSource> </InlineEquation>, μ) and T has zero entropy, then for all ci ∈ ℤ {0}, all integral polynomials pj with deg pj ⩾ 2, and all fi, gj ∈ L∞(X, μ), 1 ⩽ i ⩽ m and 1 ⩽ j ⩽ d<Equation ID="Equa"> <EquationSource Format="TEX">$\mathop {\lim }\limits_{N \to \infty } {1 \over N}\sum\limits_{n = 0}^{N - 1} {{f_1}} ( {{T^{{c_1}n}}x} ) \cdots {f_m}( {{T^{{c_m}n}}x} ) \cdot {g_1}( {{S^{{p_1}( n )}}x} ) \cdots {g_d}( {{S^{{p_d}( n )}}x} )$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <munder> <mrow class="MJX-TeXAtom-OP"> <mo form="prefix">lim</mo> </mrow> <mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo /> <mrow> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow> <mrow> <msup> <mi>T</mi> <mrow> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋯</mo> <mrow> <msub> <mi>f</mi> <mi>m</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mrow> <mrow> <msup> <mi>T</mi> <mrow> <mrow> <msub> <mi>c</mi> <mi>m</mi> </msub> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mrow> <mrow> <msup> <mi>S</mi> <mrow> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋯</mo> <mrow> <msub> <mi>g</mi> <mi>d</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mrow> <mrow> <msup> <mi>S</mi> <mrow> <mrow> <msub> <mi>p</mi> <mi>d</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </Equation>exists in L2(X, μ), which extends a recent result by Frantzikinakis and Host (2023). Moreover, it is shown that for an ergodic measure-preserving system (X, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\cal{X}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">X</mi> </mrow> </math></EquationSource> </InlineEquation>, μ, T), a non-linear integral polynomial p and f ∈ L∞(X, μ), the Furstenberg systems of (f(Tp(n)x))n∈ℤ are ergodic and isomorphic to direct products of infinite-step pronilsystems and Bernoulli systems for almost every x ∈ X, which answers a problem by Frantzikinakis (2022).

The polynomial Furstenberg joining and its applications

期刊论文
发表日期：

2025年4月22日

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Wen Huang,
Song Shao,
Xiangdong Ye

摘要

In this paper, a polynomial version of the Furstenberg joining is introduced and its structure is investigated. Particularly, it is shown that if all polynomials are non-linear, then almost every ergodic component of the joining is a direct product of an infinite-step pro-nilsystem and a Bernoulli system. As applications, some new convergence theorems are obtained. Particularly, it is proved that if T and S are ergodic measure-preserving transformations on a probability space (X, $\cal{X}$ $X$ , μ) and T has zero entropy, then for all c_i ∈ ℤ {0}, all integral polynomials p_j with deg p_j ⩾ 2, and all f_i, g_j ∈ L^∞(X, μ), 1 ⩽ i ⩽ m and 1 ⩽ j ⩽ d

$\mathop {\lim }\limits_{N \to \infty } {1 \over N}\sum\limits_{n = 0}^{N - 1} {{f_1}} ( {{T^{{c_1}n}}x} ) \cdots {f_m}( {{T^{{c_m}n}}x} ) \cdot {g_1}( {{S^{{p_1}( n )}}x} ) \cdots {g_d}( {{S^{{p_d}( n )}}x} )$ $lim_{N \to \infty} \frac{1}{N} \sum_{n = 0}^{N - 1} f_{1} (T^{c_{1} n} x) \dots f_{m} (T^{c_{m} n} x) \cdot g_{1} (S^{p_{1} (n)} x) \dots g_{d} (S^{p_{d} (n)} x)$

exists in L²(X, μ), which extends a recent result by Frantzikinakis and Host (2023). Moreover, it is shown that for an ergodic measure-preserving system (X, $\cal{X}$ $X$ , μ, T), a non-linear integral polynomial p and f ∈ L^∞(X, μ), the Furstenberg systems of (f(T^p(n)x))_n∈ℤ are ergodic and isomorphic to direct products of infinite-step pronilsystems and Bernoulli systems for almost every x ∈ X, which answers a problem by Frantzikinakis (2022).