--- title: 'Tsukamoto’s Approach: Local-to-Global Methods' url: https://www.emergentmind.com/topics/tsukamoto-s-approach type: topic --- # Tsukamoto’s Approach: Local-to-Global Methods Searching arXiv for the cited papers to ground the article in the literature. arxiv_search(query="2506.13091 OR 2606.13270 OR 2509.11985 OR 2307.16772 OR 2108.06308 OR 2510.08051 OR 2404.05293", max_results=10) Tsukamoto’s approach denotes a set of research methodologies associated with Tsukamoto’s work or later constructions explicitly formulated in Tsukamoto-style terms across several domains. In contemporary arXiv literature, the phrase is used in at least three technically distinct senses. In strong-field gravitational lensing, it refers to a refinement of Bozza’s strong-deflection formalism for asymptotically flat, static, spherically symmetric spacetimes, in which the logarithmic divergence of the deflection angle near the photon sphere is isolated and encoded by a small set of coefficients determined by the metric at the unstable circular null orbit [2506.13091], [2509.11985]. In topological dynamics and mean-dimension theory, it refers to a family of localization, recursive covering, and information-theoretic constructions that recast global invariants through local stable sets, nested covers, or rate-distortion-type quantities [2606.13270], [2307.16772], [2510.08051]. In musculoskeletal robotics, it designates a long-time self-body image acquisition strategy in which a robot continuously learns nonlinear relations among joint angles, muscle tensions, and muscle lengths from sensor data, while maintaining operational safety and control utility [2404.05293]. The common structural feature is methodological: each usage replaces a direct global treatment by a decomposition into local, recursively organized, or asymptotically singular components. ## 1. Strong-deflection gravitational lensing In black-hole optics, Tsukamoto’s approach is a strong-deflection-limit method for null geodesics in static, spherically symmetric spacetimes. The formalism is built around the unstable circular null orbit, or photon sphere, and the observation that the bending angle diverges logarithmically as the closest approach approaches the photon sphere [2506.13091], [2509.11985]. For the quantum-corrected Reissner–Nordström spacetime studied in "Shadow and strong-field lensing effects of quantum-corrected RN blackhole" [2506.13091], the metric is written as \[ ds^{2} = -A(r)\,dt^{2}+B(r)\,dr^{2}+C(r)\,d\Omega^{2}, \] with \[ B(r)=\frac{1}{A(r)}, \qquad C(r)=r^2, \] and, for small quantum parameter \(\mathrm a\), \[ A(r)_{\rm app}=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\mathrm{a}^2}{2r^2}+\mathcal{O}(\mathrm{a}^4). \] The parameter \(\mathrm a\) is defined by \[ \mathrm{a} = 4\sqrt{\kappa} \equiv 4\ell_p. \] The photon sphere is found from \[ \frac{d}{dr}\left[h^2(r_{\rm ph})\right]=0,\qquad h(r)\equiv \sqrt{\frac{C(r)}{A(r)}}, \] equivalently \[ C'(r_{\rm ph})A(r_{\rm ph})-C(r_{\rm ph})A'(r_{\rm ph})=0, \] yielding \[ r_{\text{ph}}=\frac{3M}{2}+\frac{\sqrt{9M^2-8Q^2+4\mathrm{a}^2}}{2}. \] The shadow radius is then computed from \[ r_{\rm sh}=\sqrt{\frac{C(r)}{A(r)}\Bigg|_{r_{\rm ph}} }. \] The lensing sector is formulated through \[ \alpha(r_0)=I(r_0)-\pi, \qquad I(r_0)=2\int_{r_0}^{\infty}\frac{dr}{\sqrt{\frac{R(r)C(r)}{B(r)}}}, \] with \[ R(r)=\frac{A(r_0)r^2}{A(r)r_0^2}-1. \] Tsukamoto’s refinement introduces \[ z\equiv 1-\frac{r_0}{r}, \] and splits the integral into divergent and regular parts, \[ I(r_0)=\int_0^1\left[F_D(z,r_0)+F_R(z,r_0)\right]dz. \] Near the photon sphere the bending angle takes the logarithmic form \[ \hat{\alpha}_{\rm str} = -\bar{a}\,\log\!\left(\frac{b_0}{b_{\rm crit}}-1\right)+\bar{b} +O\!\left[\left(\frac{b_0}{b_{\rm crit}}-1\right)\log\!\left(\frac{b_0}{b_{\rm crit}}-1\right)\right]. \] The generic coefficients are \[ \bar{a}= \sqrt{\frac{2}{2A(r_{\rm ps})-A''(r_{\rm ps})r_{\rm ps}^{2}}}, \] and \[ \bar{b}= \bar{a}\log\!\left[ r_{\rm ps}\left(\frac{2}{r_{\rm ps}^{2}-\frac{A''(r_{\rm ps})}{A(r_{\rm ps})}}\right) \right]+I_R(r_{\rm ps})-\pi. \] A closely related application appears in "Black Hole Gravitational Phenomena in Higher-Order Curvature-Scalar Gravity" [2509.11985]. There the spacetime is \[ ds^2=-A(r,\xi)\,dt^2+\frac{dr^2}{B(r,\xi)}+r^2(d\theta^2+\sin^2\theta\,d\phi^2), \] with \[ A(r,\xi)=1-\frac{2M}{r}+\frac{\xi}{r^2},\qquad B(r,\xi)=1-\frac{2M}{r}+\frac{2M\xi^{3/2}}{r^4}. \] The closest-approach impact parameter is \[ b(r_o)=\sqrt{\frac{r_o^2}{A(r_o,\xi)}}, \] the photon sphere is determined by \[ \frac{d}{dr}\left(\frac{r^2}{A(r,\xi)}\right)=0, \] and explicitly becomes \[ r_{\rm ph}=\frac{1}{2}\left(3M+\sqrt{9M^2-8\xi}\right). \] The critical impact parameter is \[ b_c=\sqrt{\frac{r_{\rm ph}^2}{A(r_{\rm ph},\xi)}}, \] and the strong-field deflection angle is written as \[ \alpha(b)= -\tilde a\,\ln\!\left(\frac{b}{b_c}-1\right)+\tilde b+\mathcal O\!\left[(b-b_c)\ln(b-b_c)\right]. \] Here \[ \tilde a = \sqrt{ \frac{2B_{\rm ph}}{ A_{\rm ph}C''_{\rm ph}-C_{\rm ph}A''_{\rm ph} } }, \] and \[ \tilde b = \tilde a \ln\!\left[ r_{\rm ph}^2\left( \frac{C''_{\rm ph}}{C_{\rm ph}}-\frac{A''_{\rm ph}}{A_{\rm ph}} \right) \right] +I_{\rm Reg}(r_{\rm ph})-\pi. \] This lensing usage of Tsukamoto’s approach is therefore not a generic name for black-hole imaging, but a specific asymptotic method: identify the photon sphere, transform the bending integral by \(z=1-r_o/r\), isolate the divergent part, and express the logarithmic blow-up through coefficients fixed by the metric at the photon sphere [2506.13091], [2509.11985]. ## 2. Optical observables and empirical bounds The practical significance of the strong-deflection method lies in its direct connection between local null-geodesic structure and observable quantities such as the critical impact parameter, shadow radius, angular image accumulation, and strong-lensing image separation [2506.13091], [2509.11985]. In the quantum-corrected Reissner–Nordström analysis, the paper states that charge \(Q\) decreases the photon-sphere radius \(r_{\rm ph}\), while the quantum correction parameter \(\mathrm a\) increases it [2506.13091]. The corresponding small-correction shadow expression is \[ R_{\rm sh} \approx 3\sqrt{3}M-\frac{\sqrt{3}Q^2}{2M}+\frac{\sqrt{3}\mathrm{a}^2}{4M}+\frac{7\sqrt{3}Q^2\mathrm{a}^2}{72M^3}. \] This makes explicit that \(Q^2\) decreases the shadow size, whereas \(\mathrm a^2\) increases it. The same paper reports that \(Q\) has a strong effect near the photon sphere, that the deflection angle changes significantly as \(Q\) varies close to the critical radius, and that far from the photon sphere the different curves converge and the charge dependence becomes weak. By contrast, \(\mathrm a\) has negligible impact near the photon sphere at present observational precision, but increasing \(\mathrm a\) leads to a slight increase in shadow radius and a slight suppression of the strong deflection angle [2506.13091]. The paper compares its shadow predictions with Event Horizon Telescope constraints and states the following ranges. For Sgr A*, \[ 4.209M \leq R_{\rm sh} \leq 5.560M, \qquad -0.364 \le \delta/M \le 0.987. \] For M87*, \[ 4.313M \leq R_{\rm sh} \leq 6.079M, \qquad \delta/M = \pm 0.883. \] Its main conclusion is that even for large charge the shadow remains within EHT observational bounds for both M87* and Sgr A* [2506.13091]. In the higher-order curvature-scalar gravity study, Tsukamoto’s coefficients are used to define standard observables for relativistic images: \[ \theta_\infty=\frac{b_c}{D_{OL}}, \qquad s \simeq \theta_\infty \exp\!\left(\frac{\tilde b-2\pi}{\tilde a}\right), \qquad r_m \simeq \exp\!\left(\frac{2\pi}{\tilde a}\right), \] with \[ r_m^{\rm (mag)} = 2.5\log_{10} r_m. \] The shadow angular diameter is \[ \Omega_{\rm sh}=\frac{2b_c}{\mathcal D}. \] Using EHT-measured angular diameters, the paper finds \[ 0\le \frac{\xi}{M^2}\lesssim 0.091 \] for M87\(^*\), and \[ 0\le \frac{\xi}{M^2}\lesssim 0.963 \] for Sgr A\(^*\) [2509.11985]. A plausible implication is that, within this usage, Tsukamoto’s approach functions as an observational reduction scheme: local data at the photon sphere are propagated through logarithmic asymptotics to quantities directly comparable with EHT shadow measurements. ## 3. Localization in metric mean dimension In dynamical systems, Tsukamoto’s approach denotes a localization principle for metric mean dimension. The central object is the pointwise \(\varepsilon\)-stable set \[ \Gamma_\varepsilon(x)=\{y\in X:\ d(gx,gy)\le \varepsilon \ \forall g\in G\}, \] which collects points that remain \(\varepsilon\)-close to \(x\) along the entire \(G\)-orbit [2606.13270]. The paper "Metric mean dimension of amenable group actions: localization and non-uniformity" [2606.13270] presents Tsukamoto’s original idea as the recovery of a global scale-sensitive invariant from the entropy of these small stable sets. For \(\mathbb Z\)- and \(\mathbb R^k\)-/\(\mathbb Z^k\)-actions, the localization formula is described in the form \[ \overline{\mathrm{mdim}_M(X)}=\limsup_{\delta\to 0}\sup_{x\in X}\frac{h_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta)}{\log(1/\delta)}, \] for every fixed \(\varepsilon>0\). The 2026 paper extends this viewpoint to actions of countable discrete amenable groups and proves, for every tempered Følner sequence \(\{F_n\}\) satisfying \(\lvert F_n\rvert/\log n\to\infty\), that \[ \overline{\mathrm{mdim}_{M}(X,\{F_n\},d) = \limsup_{\delta\to 0}\sup_{x\in X} \frac{h_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta,\{F_n\},d)}{\log \frac1\delta} \] \[ = \limsup_{\delta\to 0}\sup_{x\in X} \frac{h^{P}_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta,\{F_n\},d)}{\log \frac1\delta} = \limsup_{\delta\to 0}\sup_{x\in X} \frac{h^{B}_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta,\{F_n\},d)}{\log \frac1\delta}. \] This formulation is accompanied by equivalent upper metric mean dimension definitions for any nonempty \(Z\subset X\): \[ \overline{\mathrm{mdim}_M(Z,\{F_n\},d) = \limsup_{\delta\to 0}\frac{h_{\mathrm{top}}(Z,\delta,\{F_n\},d)}{\log(1/\delta)}, \] \[ \overline{\mathrm{mdim}_M^P(Z,\{F_n\},d) = \limsup_{\delta\to 0}\frac{h_{\mathrm{top}}^P(Z,\delta,\{F_n\},d)}{\log(1/\delta)}, \] \[ \overline{\mathrm{mdim}_M^B(Z,\{F_n\},d) = \limsup_{\delta\to 0}\frac{h_{\mathrm{top}}^B(Z,\delta,\{F_n\},d)}{\log(1/\delta)}. \] For the whole system, Theorem 5 yields \[ \overline{\mathrm{mdim}_M(X,\{F_n\},d) = \overline{\mathrm{mdim}_M^P(X,\{F_n\},d) = \overline{\mathrm{mdim}_M^B(X,\{F_n\},d), \] and also \[ \overline{\mathrm{mdim}_M(X,\{F_n\},d) = \max_{x\in X}\overline{\mathrm{mdim}_M(x,\{F_n\},d). \] The technical novelty of the amenable-group extension is the replacement of tiling arguments by Lindenstrauss’s combinatorial covering lemma. Proposition 14 states that if the local Bowen-dimensional entropy of all stable sets \(\Gamma_\eta(x)\) is bounded above by \(\lambda\), then for small \(\delta\), \[ \sup_{x\in X}N(\mathcal U^{F_n},B_\eta^{F_n}(x)) \le |\mathcal U|^{(\delta+\delta^{1/4})|F_n|}e^{\lambda |F_n|}2^{\delta |F_n|}. \] This estimate is identified as the amenable-group analogue of Bowen’s key covering estimate [2606.13270]. The same paper also establishes a limitation of the localization principle. It defines \[ \overline D_{\mathrm{int}}(X,\varepsilon) = \limsup_{\delta\to 0}\sup_{x\in X} \frac{h_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta,\{F_n\},d)}{\log(1/\delta)}, \] \[ \overline D_{\mathrm{ext}}(X,\varepsilon) = \sup_{x\in X}\limsup_{\delta\to 0} \frac{h_{\mathrm{top}}(\Gamma_\varepsilon(x),\delta,\{F_n\},d)}{\log(1/\delta)}, \] with analogous lower quantities. Theorem 7 gives a \(\mathbb Z\)-subshift \(X\) such that for every \(\varepsilon\in(0,1)\), \[ \overline D_{\mathrm{int}}(X,\varepsilon)=\underline D_{\mathrm{int}}(X,\varepsilon)=1, \qquad \overline D_{\mathrm{ext}}(X,\varepsilon)=\underline D_{\mathrm{ext}}(X,\varepsilon)=0. \] Thus the supremum and small-scale limit cannot generally be interchanged [2606.13270]. This establishes a characteristic feature of Tsukamoto’s localization approach: the global invariant is indeed recoverable from local stable sets, but not uniformly across phase space. ## 4. Recursive coverings and weighted topological pressure A second dynamical-systems usage of Tsukamoto’s approach appears in weighted entropy and weighted pressure. The paper "Weighted topological pressure revisited" [2307.16772] describes Tsukamoto’s 2022 contribution as a redefinition of weighted pressure in a more “nested covering” style for the case of two systems, and then generalizes that idea to an arbitrary finite chain of factor maps. The setting is a sequence \[ (X_1,T_1)\xrightarrow{\pi_1}(X_2,T_2)\xrightarrow{\pi_2}\cdots\xrightarrow{\pi_{r-1}}(X_r,T_r), \] together with \[ \boldsymbol a=(a_1,\dots,a_{r-1})\in[0,1]^{r-1}. \] From \(\boldsymbol a\) one forms \[ \boldsymbol w_{\boldsymbol a}=(w_1,\dots,w_r) \] via \[ \begin{cases} w_1=a_1a_2\cdots a_{r-1},\\ w_2=(1-a_1)a_2\cdots a_{r-1},\\ w_3=(1-a_2)a_3\cdots a_{r-1},\\ \vdots\\ w_{r-1}=(1-a_{r-2})a_{r-1},\\ w_r=1-a_{r-1}. \end{cases} \] These weights sum to \(1\) [2307.16772]. The new weighted entropy is defined recursively. With \[ d^{(i)}_N(x,y)=\max_{0\le n\sqrt{k}/2\), \[ B_r(V)=\{u\in \mathbb{Z}^k:\ |u-w|