arXiv subjects

Filippo Ciavattini

Publications and source records attributed to Filippo Ciavattini.

3 recordsLinked to original sources

arXiv2026

Emergent order spectrum for transitive homeomorphisms

The Emergent Order Spectrum $\Omega(x,y)$ is a topological invariant of dynamical systems providing order-types induced by the limit order of order-compatible nested $\varepsilon_n$-chains (with $\varepsilon_n\to 0$) from $x$ to $y$. In this paper, we investigate how rich these spectra can be under natural dynamical hypotheses. For a transitive homeomorphism $f$ of a compact metric space $X$ without isolated points and of cardinality $\mathfrak{c}$, we show that the global spectrum $\Omega_f(X^2)$ is universal at the countable scattered level: every countable scattered order-type together with the order-type of the rationals appears in $\Omega_f(X^2)$. More precisely, there exists a comeagre subset $M\subseteq X^2$ such that, for every $(x,y)\in M$, the individual spectrum $\Omega_f(x,y)$ already realizes all countably infinite scattered order-types; moreover, the order-type of the rationals belongs to $\Omega_f(x,y)$ for every pair $(x,y)\in X^2$.

math.DS↗

arXiv2025

Nowhere dense competing holes in open dynamical systems

Let $\mathcal{M}$ be a compact metric space with no isolated points, and $f:\mathcal{M}\longrightarrow\mathcal{M}$ a homeomorphism. Consider a sequence of shrinking open balls $\{B^i_n\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$ with centers $\{p_i\}_{i=1}^\infty\subseteq\mathcal{M}$ and radii $\{ρ^i_n\}_{n=1}^\infty$. For every point $x\in\mathcal{M}$ and $n\in\mathbb{N}$, consider which ball the trajectory $\{x,f(x),f^2(x),\dots\}$ of the point first visits. We find that whenever the closure of $\{p_i\}_{i=1}^\infty$ is nowhere dense, and with very minor restrictions on $\{ρ_n^i\}_{n\in\mathbb{N}}^{i\in\mathbb{N}}$, the typical trajectory $\{f^k(x)\}_{k=0}^\infty$ will first visit, for each $i$, the ball $B^i_n$, for infinitely many $n$. This is never the case, should $\{p_i\}_{i=1}^\infty$ be somewhere dense. Keywords: Open Dynamical System, Topological Dynamics, Transitive Homeomorphism, Baire category. MSC2020: 37B05, 37B20, 18F60, 54E52.

math.DS↗

arXiv2025

Adding machines and open dynamical systems

Let $f:\mathcal{M}\rightarrow\mathcal{M}$ be a continuous map defined on a compact metric space $\mathcal{M}$. An open dynamical system introduces disjoint open balls centered at points in $\mathcal{M}$, and considers the trajectories of points from $\mathcal{M}$, and the balls that they visit first. As the balls in question are allowed to shrink, a point is considered indecisive if its trajectory changes infinitely many times the ball first visited. Here, we let $\mathcal{M}$ be an adding machine, a simple system and a solenoidal system. In each case, we show that the set of points which generate indecisive trajectories is residual. Keywords: Open dynamical system, Adding machine, Solenoidal system, Baire category

math.DS↗