Overview
This study addresses the pointwise convergence of double ergodic averages. Specifically, it examines a particular class of these averages constructed along non-polynomial sequences. The central finding demonstrates the existence of a limit for such averages under specified conditions.
Research Context
The investigation centers on ergodic theory, a branch of mathematics concerned with dynamical systems and measure theory. Within this field, the behavior of ergodic averages, particularly their convergence properties, constitutes a fundamental area of inquiry. The context of this research involves measure preserving systems, which are foundational structures in ergodic theory for modeling systems that evolve over time while conserving a measure.
Approach
The research establishes a result for a specific mathematical expression representing double ergodic averages. The approach involves defining a constant $c$ within the open interval $(1,2)$, i.e., $c \in (1,2)$. Additionally, two non-zero real numbers, $\alpha$ and $\beta$, are fixed. The framework for the analysis is a general measure preserving system, denoted $(X,\mathcal{X},\mu,T)$, where $X$ is a set, $\mathcal{X}$ is a $\sigma$-algebra on $X$, $\mu$ is a measure, and $T$ is a measure-preserving transformation. Functions $f$ and $g$ are considered from the space $L^{\infty}(\mu)$, which comprises essentially bounded measurable functions.
The specific limit under consideration is given by the formula:
$$ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\lfloor \alpha n^c \rfloor}x)g(T^{\lfloor \beta n^c \rfloor}x) $$
This expression represents an average over $N$ terms, where each term involves the functions $f$ and $g$ evaluated at points obtained by applying the transformation $T$ a number of times. The number of applications of $T$ is determined by the sequences $\lfloor \alpha n^c \rfloor$ and $\lfloor \beta n^c \rfloor$, which are non-polynomial sequences due to the exponent $c$ not being an integer and the floor function.
Findings
The principal finding of the research is that, under the specified conditions (fixed $c\in (1,2)$, non-zero real numbers $\alpha$ and $\beta$, and any measure preserving system $(X,\mathcal{X},\mu,T)$ with $f,g\in L^{\infty}(\mu)$), the limit for the double ergodic average
$$ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\lfloor \alpha n^c \rfloor}x)g(T^{\lfloor \beta n^c \rfloor}x) $$ exists for $\mu$-a.e. $x\in X$. This result indicates that for almost every point $x$ in the space $X$, the average converges to a specific value as $N$ tends to infinity.
Why This Matters
The explicit text does not contain information on why this matters, potential applications, or limitations.