A Markus–Yamabe/La Salle problem for discrete dynamical systems

Definitions and mathematical statement

Let $F:\mathbb R^2\to\mathbb R^2$ be smooth and suppose $F(a)=a$ for some $a\in\mathbb R^2$. For a real matrix $M=(m_{ij})$, write $|M|=(|m_{ij}|)$ for entrywise absolute value, and let $\rho(M)$ be its spectral radius, the greatest modulus of its complex eigenvalues. Assume \[ \rho\left(|DF(x)|\right)<1\qquad\text{for every }x\in\mathbb R^2. \] The question is whether $a$ must be globally asymptotically stable for the discrete dynamical system $x_{j+1}=F(x_j)$.

Here global asymptotic stability requires both \[ \lim_{j\to\infty}F^j(x)=a\quad\text{for all }x\in\mathbb R^2 \] and Lyapunov stability: for every $\varepsilon>0$ there is $\delta>0$ such that $\|x-a\|<\delta$ implies $\|F^j(x)-a\|<\varepsilon$ for all integers $j\ge0$. All norms on $\mathbb R^2$ give the same qualitative definition.

The absolute value in the hypothesis is essential: it is taken before computing the spectral radius, not afterwards. Nor is the condition a uniform contraction estimate in a single chosen norm. The permitted spectral radii may approach one as $x$ goes to infinity, and no global bound on $DF$ is assumed. Thus a proof that invokes a fixed contraction constant must first derive it, rather than silently adding it as a hypothesis. The fixed point is assumed to exist; its uniqueness would be part of the asserted conclusion.

Short English statement

Does pointwise spectral-radius control of the entrywise absolute Jacobian force global convergence to the fixed point?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700020 (AMR-046-0020), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 20: A Markus–Yamabe/La Salle problem for discrete dynamical systems. Original source indexed by AMR-046-0020. https://arxiv.org/abs/2012.02524.