José Pablo Santander

arXiv2026

On the topological dynamical system of tilings

The compactness of the tiling space and the continuity of the action of the Euclidean space by translation, as well as the minimality of this action for repetitive tilings, are among the most frequently mentioned foundational facts in the dynamical theory of aperiodic order. Nonetheless, despite their classical status, a comprehensive treatise with complete and detailed proofs, to the best of our knowledge, is not available in the literature: existing arguments are typically sketched, deferred to other sources, or rely on unstated results or hypotheses. The purpose of this manuscript is to give a self-contained and pedagogical treatment of these foundations, providing complete and carefully explained proofs. We hope that this exposition will provide a useful reference and contribute to a clearer and more accessible understanding of the fundamentals of the theory of tiling dynamical systems.

math.DS↗

arXiv2026

Duality for the $Γ$-convergence of convex functions

We extend the duality principle for the $Γ$-convergence of convex lower semicontinuous functions, which was previously established only in separable reflexive Banach spaces, to the broader class of weakly compactly generated (WCG) Banach spaces, addressing a question of Fitzpatrick and Lewis. Under the same classical hypothesis of equicoercivity, we show that $Γ$-convergence in the norm topology is equivalent to $Γ$-convergence of the Fenchel conjugates in the weak$^\ast$ topology. We further prove that this duality is equivalent to the graphical convergence of the associated subdifferentials with respect to the product topology given by the norm on the primal space and the weak$^\ast$ topology on the dual. The WCG setting encompasses all separable and all reflexive Banach spaces separately, i.e, separable spaces without reflexivity assumptions and reflexive spaces without separability assumptions, as well as important non-reflexive spaces which may fail to be separable, such as $L^1(μ)$ for an arbitrary $σ$-finite measure. As an application, we derive dual characterizations of the $Γ$-convergence of convex integral functionals on $L^p$ spaces ($1\leq p<\infty $).

math.FA↗