Hölder Selections under Stieltjes Clocks: Uniform Disconnectedness and Jump Dominance
Abstract
It is known that, for ordinary Hölder multifunctions with 0 < α < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < α < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.
Related
- Selections of Set-Valued Maps under Stieltjes Clocks: Regularity, Variation, and Atomic Structure
- Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock
- Tight Lower Bounds under Matched High-Order Hölder Smoothness and Uniform Convexity
- Disconnectedness of the Hilbert Schemes of E6/P6
- Disconnectedness of stability manifolds and their geometric loci, and separation of braid orbits of semiorthogonal decompositions
BibTeX
@misc{ilter2026hölder,
title = {{Hölder Selections under Stieltjes Clocks: Uniform Disconnectedness and Jump Dominance}},
author = {Serkan İlter and Hülya Duru and Arkady Kitover and Mehmet Orhon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20706},
eprint = {2609.20706},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20706},
}
RIS
TY - UNPB
TI - Hölder Selections under Stieltjes Clocks: Uniform Disconnectedness and Jump Dominance
AU - Serkan İlter
AU - Hülya Duru
AU - Arkady Kitover
AU - Mehmet Orhon
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20706
UR - https://arxiv.org/abs/2609.20706
AB - It is known that, for ordinary Hölder multifunctions with 0 < α < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < α < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.
ER -