[Paper Review] Simple Spectrum for Tensor Products of Mixing Map Powers

V. V. Ryzhikov|arXiv (Cornell University)|Jul 24, 2011

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper constructs a mixing rank-one transformation $T$ such that the infinite tensor product $T \otimes T^2 \otimes T^3 \otimes \cdots$ has simple spectrum. Using stochastic rank-one constructions with tailored spacer sequences, the authors ensure specific weak limits for powers of $T$, enabling spectral disjointness and cyclic vector generation in the tensor product space, thereby establishing simple spectrum via inductive spectral arguments and weak convergence techniques.

ABSTRACT

We prove the existence of a mixing rank one map $T$ such that the product $T\otimes T^2\otimes T^3\otimes...$ has simple spectrum. This result is used in S.V. Thikhonov's proof of the existence of mixing transformation with homogeneous spectrum of multiplicity $m>2$.

Motivation & Objective

• To construct a mixing measure-preserving transformation $T$ such that the infinite tensor product $T \otimes T^2 \otimes T^3 \otimes \cdots$ has simple spectrum.
• To extend spectral techniques from finite products to infinite tensor products of powers of a single transformation.
• To provide a constructive method using stochastic rank-one constructions with controlled spacer sequences to achieve desired spectral behavior in tensor products.
• To resolve a spectral problem in ergodic theory by establishing simple spectrum for infinite tensor products of mixing map powers.

Proposed method

• Use stochastic rank-one constructions with spacer sequences designed to produce specific weak limits for powers of $T$, particularly $T^{km_j} \to a_k I + (1 - a_k)\Theta$ for $k \neq s$.
• Introduce three-part spacer arrays: an $a$-part for constant spacers, a $b$-part with periodic $nH_j - 1$ spacers to generate $T$-like behavior, and a $c$-part with increasing spacers to induce mixing.
• Employ weak operator convergence $\approx_w$ to analyze the asymptotic behavior of $T^{km_j}$, ensuring convergence to combinations of identity, shift, and projection onto constants.
• Use induction and spectral disjointness from $(s,n)$-properties to show that the cyclic subspace generated by $f^{\otimes n}$ is dense in $L^2 \otimes \cdots \otimes L^2$, implying simple spectrum.
• Leverage the fact that different products $T^{n_1} \otimes \cdots \otimes T^{n_k}$ are spectrally disjoint due to distinct weak limits, ensuring no spectral overlap.
• Apply a limiting argument: construct a sequence of $c_p$-partially mixing transformations $T_p$ with simple spectrum for $T_p \otimes T_p^2 \otimes \cdots$, then take $T_p \to T$ as $p \to \infty$ to obtain a mixing limit with the desired spectral property.

Experimental results

Research questions

• RQ1 Can a mixing transformation $T$ be constructed such that the infinite tensor product $T \otimes T^2 \otimes T^3 \otimes \cdots$ has simple spectrum?
• RQ2 What conditions on the powers $T^k$ ensure spectral disjointness in the tensor product space?
• RQ3 How can stochastic rank-one constructions be engineered to produce specific weak limits $T^{km_j} \to a_k I + (1 - a_k)\Theta$ for $k \neq s$, while $T^{sm_j} \to \Theta$?
• RQ4 Is it possible to extend the spectral simplicity of finite tensor products to infinite products via a limiting construction?
• RQ5 Can the spectral behavior of flows be similarly controlled to ensure simple spectrum in products of their exponentials?

Key findings

• A mixing rank-one transformation $T$ exists such that the infinite tensor product $T \otimes T^2 \otimes T^3 \otimes \cdots$ has simple spectrum.
• The construction uses a stochastic rank-one framework with three-part spacer sequences to achieve $T^{km_j} \to a_k I + (1 - a_k)\Theta$ for $k \neq s$ and $T^{sm_j} \to \Theta$, ensuring spectral disjointness.
• For $k = 1, 2, \dots, n-1$, the powers $T^{km_j}$ converge weakly to $a_k I + (1 - a_k)\Theta$, while $T^{nm_j} \to aI + bT + c\Theta$, enabling inductive spectral control.
• The cyclic subspace generated by $f^{\otimes n}$ is dense in $H^{\otimes n}$, proving that $T \otimes T^2 \otimes \cdots \otimes T^n$ has simple spectrum.
• The limit transformation $T$ obtained as $T_p \to T$ is mixing and inherits the spectral simplicity of the finite products via standard approximation techniques.
• The result supports a conjecture on the existence of a mixing flow $T_t$ such that all products $T_{t_1} \otimes T_{t_2} \otimes \cdots$ have simple spectrum, with a potential spectral criterion involving weak limits of $T_{t_i m_j}$.