{"paper":{"title":"A shift map with a discontinuous entropy function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Christian Wolf","submitted_at":"2018-03-06T22:15:10Z","abstract_excerpt":"Let $f:X\\to X$ be a continuous map on a compact metric space with finite topological entropy. Further, we assume that the entropy map $\\mu\\mapsto h_\\mu(f)$ is upper semi-continuous. It is well-known that this implies the continuity of the localized entropy function of a given continuous potential $\\phi:X\\to R$. In this note we show that this result does not carry over to the case of higher-dimensional potentials $\\Phi:X\\to R^m$. Namely, we construct for a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\\Phi$ with a discontinuous localized entropy function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.02440","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}