arXiv subjects

Marco Farotti

Publications and source records attributed to Marco Farotti.

5 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

Chains without regularity

We study chain-recurrence and chain-transitivity in compact dynamical systems without any regularity assumptions on the map. We prove that every compact system has a chain-recurrent point and a closed, invariant, chain-transitive subsystem. The proofs do not rely on the Axiom of Choice.

math.DS↗

arXiv2025

Transfinite Topological Dynamics

We present a canonical extension of topological dynamics to transfinite iterations, which makes precise the idea of dynamical phenomena stabilizing at different time-scales. Specifically, consider a sequence of self-maps $F=\{f_n\}$ of a compact metric space $X$. If $F$ is finitely convergent, i.e. $f_n(x)=f(x)$ for $n>N(x)$, the $f_n$-orbits exhibit an emergent poset structure. A maximal initial segment of this poset is isomorphic to a countable ordinal $\ge\omega$. The construction is canonical: every finitely convergent sequence induces, at each point, a unique maximal transfinite orbit that is independent of any finite initial segment of the sequence and invariant under step-by-step conjugacy at each $n$. For $\lambda$ a countable limit ordinal, we study orbits, recurrence, limit sets and attractors at level $\lambda$, and the interplay of different ordinal levels. Moreover, we introduce the natural notion of transfinite conjugacy, that refines conjugacy of limit maps alone but is strictly weaker than step-by-step conjugacy. We describe a family of invariants of transfinite conjugacy that detect recurrence and attraction phenomena at each ordinal level. Particularizing to $\lambda=\omega$ recovers (and in some cases refines) classical results of topological dynamics.

math.DS↗

arXiv2024

Dynamical properties of critical exponent functions

In the last years the attention towards topological dynamical properties of highly discontinuous maps has increased significantly. In [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Mathématique, 360(G4), 315-332, 2022], a class of densely discontinuous interval maps, called "critical exponent maps", was introduced. These maps are defined through the word-combinatorics concept of critical exponent applied to the binary expansion of reals and show highly chaotically properties as well as some challenging problems. In this paper we identify an error in the proof of Theorem 7 in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Mathématique, 360(G4), 315-332, 2022], a purely combinatorial result which in fact does not hold. We show that most of the results in [D.Corona, A. Della Corte. The critical exponent functions. Comptes Rendus Mathématique, 360(G4), 315-332, 2022], obtained there through Theorem 7, can be recovered. Moreover, we propose as a conjecture a weaker form of Theorem 7.

math.DS↗

arXiv2021

Dynamics of interval maps generated by erasing substitutions

We study discontinuous interval maps generated by the action of erasing block substitutions on the binary expansion. After establishing some general properties of these maps, we categorize erasing block substitutions in a hierarchy of classes displaying progressively stronger erasing character. We investigate how this affects the dynamics of the corresponding interval maps, showing that the richest dynamical behavior (Devaney and Li-Yorke chaos, infinite topological entropy) is achieved at a precise step in this hierarchy, which we name completely erasing substitutions. KEYWORDS: Topological dynamics, Erasing substitutions; Devaney chaos; Li-Yorke chaos; Topological entropy.

math.DS↗