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Oscillations of BV measures on nested fractals

Patricia Alonso Ruiz, Fabrice Baudoin

Abstract

Motivated by recent developments in the theory of bounded variation functions on nested fractals, this paper studies the exact asymptotics of functionals related to the total variation measure associated with unions of $n$-complexes. The oscillatory behavior observed implies the non-uniqueness of BV measures in this setting.

Oscillations of BV measures on nested fractals

Abstract

Motivated by recent developments in the theory of bounded variation functions on nested fractals, this paper studies the exact asymptotics of functionals related to the total variation measure associated with unions of -complexes. The oscillatory behavior observed implies the non-uniqueness of BV measures in this setting.
Paper Structure (14 sections, 17 theorems, 104 equations, 4 figures)

This paper contains 14 sections, 17 theorems, 104 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations and set up
  3. Compact nested fractals
  4. Unbounded nested fractals
  5. Brownian motion and heat kernels
  6. Korevaar-Schoen-Sobolev and BV spaces on fractals
  7. A renewal lemma
  8. Heat semigroup functional
  9. Preliminary lemmas
  10. Proof of Theorem \ref{['T:liminf_Ucells_heat']}
  11. Korevaar-Schoen functional
  12. Scaling and localization
  13. Finitely ramified nested fractals
  14. Non-existence of the limit

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be an unbounded nested fractals as described in Section S:setup, with length scaling factor $L$, Hausdorff dimension $d_h$ and walk dimension $d_w$. There exist positive and bounded periodic functions $\Phi$ and $\Psi$ with period $L^{-d_w}$, respectively $L^{-1}$, such that for any Here, $|\partial U|$ equals the number of points in the boundary of $U$. In the case of the unbound

Figures (4)

  • Figure 1: Unions of cells in the Vicsek set (left) and in the Sierpinski gasket (right)
  • Figure 2: The Vicsek set (left) and the Sierpinski gasket (right)
  • Figure 3: The set $U$ in red. The two blue cells correspond to $K^*_w$.
  • Figure 4: Intersections of $n$-cells with balls of radius $r_n$ (left) and $r'_n$ (right) in the Sierpinski gasket (above) and the Vicsek set (below).

Theorems & Definitions (38)

  • Theorem 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Definition 2.5: Heat semigroup Besov classes
  • Definition 2.6: BV class
  • Theorem 2.7: Heat semigroup characterization of BV functions
  • Lemma 2.8: Renewal lemma Ham11
  • ...and 28 more