Derivations and Dirichlet forms on fractals

Abstract

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including nonself-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

Densely defined non‐closable curl on carpet‐like metric measure spaces

“…We consider the Hilbert space of L 2 ‐ differential 1‐forms associated with and the corresponding first order derivation as introduced in and studied in . This is a generalized L 2 ‐theory of 1‐forms.…”

“…We consider the space spanned by tensors , by they may be viewed as elements of . In order to follow the notation of , the notation used here deviates slightly from the one in [, Section 8.1], but structurally the definitions are the same. Right and left actions of on can be defined as the linear extensions of The definition yields a linear operator that satisfies a product rule , The space …”

“…As is fixed, we write  → Γ( , )( ) for the densities Γ( , ) . We consider the Hilbert space (, ⟨⋅, ⋅⟩  ) of 2differential 1-forms associated with (,  ) and the corresponding first order derivation 0 ∶  →  as introduced in [14] and studied in [15,33,34,36,38]. This is a generalized 2 -theory of 1-forms.…”

“…Our paper is a part of a broader program that aims to connect research on derivatives on fractals ( [8,9,[13][14][15][25][26][27][28]31,33,35,[37][38][39][40][41]46,51,54] and references therein) and on more general regular Dirichlet spaces [29,30,34] with classical and geometric analysis on metric measure spaces ( [6,10,12,[21][22][23][24][42][43][44][45]50] and references therein). In our previous article [36] we showed that on certain topologically one-dimensional spaces with a strongly local regular Dirichlet form one can prove a natural version of the Hodge theorem for 1-forms defined in 2 -sense: the set of harmonic 1-forms is dense in the orthogonal complement of the exact 1-forms.…”

“…In particular the reader can consult the papers [7,10,[42][43][44][45] and references therein. Our current paper is a step in a long-term program (see [29][30][31][32][33][34][35][36][37][38]) to develop parts of differential geometry and their applications to mathematical physics (see [3][4][5]) for spaces that carry diffusion processes but no other smooth structure. Our approach is somewhat complementary to the celebrated works [12,21,22] because, although our spaces are metrizable, we do not use any particular metric in an essential way, and we do not use functional inequalities.…”