Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2104.08900

Pressures for multi-potentials in semigroup dynamics

Eugen Mihailescu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study several notions of topological pressure and capacities for multi-potentials $Φ\in \mathcal C(X;\mathbb R)^m$, with respect to finitely generated continuous semigroups $G$ on a compact metric space $X$. We introduce the amalgamated pressure, the condensed pressure, the trajectory pressure, the exhaustive pressure, and the respective capacities on non-compact sets $Y$, for multi-potentials $Φ$. This is done by using Carathéodory-Pesin structures. Several properties of these types of pressure, and relations between them are explored. The inverse limit of the semigroup and its relations to the above pressures are studied. These notions can be used to classify semigroup actions. We introduce a notion of measure-theoretic amalgamated entropy, and prove a Partial Variational Principle for the amalgamated pressure. Local amalgamated entropies and local exhaustive entropies are introduced for probability measures on X, and we show they provide estimates for the amalgamated and the exhaustive entropies of non-compact sets. Also we find a bound for the local exhaustive entropy for marginal measures on $X$. We apply the amalgamated pressure of the unstable multi-potential to estimate the dimensions of slices in $G$-invariant saddle-type sets.

Explore related subjects

math.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eugen Mihailescu. 2021-07-19. Pressures for multi-potentials in semigroup dynamics. https://arxiv.org/abs/2104.08900

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

KEEP EXPLORING

Related papers

arXiv2026

Equidistribution of saddle periodic points for Hénon-like maps

We prove that under a natural assumption on the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of the results of Bedford-Lyubich-Smillie, Dujardin, and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map.

math.DS↗
arXiv2026

On dissonance and orthogonal projections of self-conformal measures

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. We establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. In addition, we show that $\dim μ\circπ^{-1} = \min\{ k, \dim μ\}$ for every ortohogonal projection $π:\mathbb{R}^d\to\mathbb{R}^k$, $0<k<d$, when either $d=2$ and $μ$ is not self-similar and not supported on a line, or $d\geq 3$ and $μ$ is totally non-linear and not supported on a smooth hypersurface.

math.DS↗
arXiv2026

Equation-Free Screening of Mittag-Leffler-Compatible Dynamics from Scalar Time Series via kNN Multi-Horizon Profiles

Fractional models provide a natural description of systems with memory, but a noninteger derivative should not be introduced solely because a time series is curved or slowly relaxing. We develop an equation-free preliminary screening framework that asks whether a scalar time series produces a multi-horizon k-nearest-neighbor (kNN) profile more compatible with Mittag-Leffler-type behavior than with selected conventional alternatives. In an ideal matched Caputo-relaxation benchmark, the complete generation-kNN-profile-model-comparison pipeline reproduces the expected Mittag-Leffler geometry and recovers the generating order to within approximately $10^{-3}$; this is interpreted as controlled calibration rather than as general fractional-order identification. Under 3% trajectory-specific observational noise, the held-out Mittag-Leffler preference is most consistent when the generating dynamics are well separated from the integer-order limit and becomes progressively less decisive as $α\rightarrow1$. The fitted order $α_{\mathrm{fit}}$, however, shows substantially larger realization-to-realization variability. Thus, relative model compatibility is more robust than single-realization order estimation in the present noisy benchmark. Noise-free nonfractional controls show a separate limitation of specificity: a stretched exponential can generate a strongly Mittag-Leffler-compatible profile, whereas inclusion of the generating rational/Hill family recovers that family and its parameters to numerical precision in the matched setting. A positive Mittag-Leffler-versus-exponential screen therefore does not uniquely establish fractional origin. A fractional chaotic system is treated only as an exploratory extension: the Mittag-Leffler growth family gives lower finite-window RMSE than exponential and logistic/saturating alternatives over the detected pre-transition interval.

math.DS↗