Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.03287

Homological reduction and entropy of discontinuous interval maps

Henrique M. Oliveira
View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a homological approach to the topological entropy of discontinuous interval maps with arbitrary branch orientations. Relations among the one-sided images of discontinuities reduce the kneading determinant, while symmetries yield further simplifications. We determine the smallest possible dimension of this reduction for each orientation pattern and show that this minimum can coexist with maximal entropy. The general theory is applied to maps with increasing or decreasing branches, extending earlier results for bimodal maps and yielding exact formulas for entropy and periodic-orbit counts.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Henrique M. Oliveira. 2026-11-02. Homological reduction and entropy of discontinuous interval maps. https://arxiv.org/abs/2611.03287

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hilbert's 16th problem through Morse theory

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound turns out to be a polynomial of degree $4$ in $n$. More specifically, if $H(n)$ indicates the maximum number of limit cycles among planar, differential, polynomial systems of degree $n$, then \begin{gather} H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{43}2n^2-\dfrac{37}2n+7\,\,\,\, \mbox{if $n$ is even, and} \nonumber H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{41}2n^2-\dfrac{33}2n+6\,\,\,\, \mbox{if $n$ is odd}.\nonumber \end{gather} For quadratic systems, we find $H(2)=4$. Our proof is entirely variational and utilizes in a fundamental way tools and facts from Morse theory, to the point that no particular expertise in dynamical systems is necessary or required. It is particularly interesting to compare our results with the ones for the same problem recently announced and made public by OpenAI as part of the solution of hundreds of important mathematical problems.

math.DS↗
arXiv2026

On the Dynamics of Invariant Graphs for Dissipative Twist Maps

We study the persistence and internal dynamics of invariant graphs for dissipative twist maps. Explicit $C^1$ persistence and parameter estimates allow every real rotation number to be realized by varying either translation parameter, while arbitrarily small zero-mean analytic perturbations at fixed irrational horizontal parameters can yield analytic graphs with dynamics conjugate to rational rotations. With a fixed Brjuno horizontal parameter and fixed-strip analyticity, normally attracting $C^r$ graphs persist under perturbations of size $(1-λ)^γ$, for any $γ>0$ and prescribed finite $r$, as $λ\to1^-$. At the critical Lipschitz bound, the invariant graphs can carry Denjoy dynamics and have countably many nondifferentiable points, with perturbations smooth outside a null Cantor set and tending to zero in every Hölder norm of order less than one.

math.DS↗
arXiv2026

Non-stationary normal coordinates on neighborhoods of Pesin stable manifolds

We construct non-stationary normal coordinates in a neighborhood of Pesin stable manifolds \cite{Pesin}. This construction is a natural extension, via jets in the normal directions, of the non-stationary normal coordinates on stable manifolds introduced by Guysinsky and Katok. We emphasize that this extension provides a useful framework for describing the fractal geometric structures of stable and unstable foliations in smooth dynamical systems.

math.DS↗