Spectral Construction of Optimal Weighted Wasserstein Metrics for Random Logistic-Type Maps
Year: 2026
Type: book-chapter
Abstract: Random iterated function systems of one-dimensional maps on a compact interval $$X\subset \mathbb {R}$$ X ⊂ R often fail to be globally contractive in any fixed metric, even when sample Lyapunov exponents are negative. A classical workaround is to twist the geometry by a positive weight w and require a weighted-derivative condition of the form $$ \sum _{k=1}^m p_k \left| f_k'(x)\,\frac{w(x)}{w(f_k(x))}\right| \le r<1, \qquad \forall x\in X, $$ ∑ k = 1 m p k f k ′ ( x ) w ( x ) w ( f k ( x ) ) ≤ r < 1 , ∀ x ∈ X , which ensures that the associated Markov operator is a strict contraction in a weighted Wasserstein–Kantorovich metric $$d_W^{(w)}$$ d W ( w ) . In this paper, we replace ad hoc constructions of w by a spectral theory for the derivative transfer operator $$ (\mathscr {L}\varphi )(x) = \sum _{k=1}^m p_k\,|f_k'(x)|\,\varphi (f_k(x)), \qquad \varphi \in C(X). $$ ( L φ ) ( x ) = ∑ k = 1 m p k | f k ′ ( x ) | φ ( f k ( x ) ) , φ ∈ C ( X ) . Our main result proves that the optimal contraction constant, minimised over all continuous weights $$w>0$$ w > 0 , coincides with the spectral radius $$\rho (\mathscr {L})$$
Author: Ramen Ghosh
Institution: Atlantic Technological University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W7160875554