arXiv subjects

Boris Petković

Publications and source records attributed to Boris Petković.

6 recordsLinked to original sources

arXiv2026

Topological centralizers of perturbations of ergodic toral automorphisms with two dimensional center

Let $A$ be an ergodic linear automorphism of the torus $\T^N$ whose center space has dimension two. We prove a dichotomy result that for every $f \in \mathcal{U}$, where $\mathcal{U}$ is the $C^{22}$ neighborhood of $A$ inside volume preserving diffeomorphisms on $\T^N$, exactly one of two alternatives holds. Either $f$ has the accessibility property, or $f$ is topologically conjugate to $A$ by a homeomorphism homotopic to the identity which simultaneously conjugates every homeomorphism commuting with $f$ to an affine automorphism of the torus. No irreducibility of the characteristic polynomial of $A$ is assumed. The alternatives exclude each other because a partially hyperbolic diffeomorphism topologically conjugate to $A$ is never accessible, so within $\mathcal{U}$ the failure of accessibility is equivalent to topological conjugacy to $A$. In the second case the centralizer of $f$ in the group of homeomorphisms of the torus is isomorphic to the group of affine transformations commuting with $A$. When the characteristic polynomial of $A$ is irreducible we show that this group is a finite extension of the unit group of an order in the number field generated by an eigenvalue of $A$, hence virtually free abelian of rank $r_1 + r_2 - 1$. The characteristic polynomial is allowed to be reducible, in which case the same description holds with the linear parts ranging over the centralizer of $A$ in $\GL(N,\Z)$. Our method uses an elementary density property of the projection of the lattice to the center space, which replaces irreducibility and which we establish for every ergodic automorphism with no restriction on the dimension of the center space.

math.DS↗

arXiv2026

An exponent-$s$ dynamical Borel-Cantelli lemma and the waiting time problem

Galatolo and Kim proved that the dynamical Borel-Cantelli property for decreasing sequences of balls is tightly connected with the waiting time problem. In systems where all such sequences are Borel-Cantelli, the time needed to enter a small ball $B$ for the first time scales as $μ(B)^{-1}$, and conversely, waiting time estimates yield Borel-Cantelli results for sequences of balls whose radii decrease in a controlled way. We extend this correspondence to the exponent-$s$ setting introduced by Tseng. For $s\ge 1$, the $s$-exponent monotone shrinking target property ($s$MSTP) requires the Borel-Cantelli conclusion only for decreasing sequences of centered balls satisfying the stronger divergence condition $\sum_nμ(B_n)^s=\infty$. We prove that $s$MSTP forces the lower waiting time exponent, measured on the scale of $-\logμ(B(y,r))$, to lie in the interval $[1,s]$ almost everywhere. That a quantitative ($s$-strong) form of the property bounds the upper exponent by $s$ and that, conversely, an exponent-$s$ waiting time estimate implies the Borel-Cantelli property for decreasing sequences of centered balls whose radii obey the calibrated decay condition matching the critical divergence exponent $s$. We also obtain the corresponding quantitative orbit approximation statement $\liminf_n n^β\,d(T^nx,y)=0$ for $β<1/(s\,\underline{d}_μ(y))$, show that the universal lower bound with exponent $1$ pins the theory to $s\ge 1$, and discuss sharpness on circle rotations, where by results of Kurzweil, Kim-Seo and Tseng the picture is governed by the Diophantine type of the rotation number.

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arXiv2026

Classification of some cohomologically $C^0$-stable continuous group actions on metric spaces

After Katok, a homeomorphism $f\colon M\to M$ of a compact metric space is said to be cohomologically $C^0$-stable if its space of real $C^0$-coboundaries is closed in $C^0(M)$. Kocsard proved that this is the case if and only if $f$ is periodic. We extend the classification to actions of arbitrary finitely generated groups: an action $α: G \to Homeo(M)$ is cohomologically $C^0$ - stable if and only if the image $α(G)$ is a finite group. In particular this settles the case of $\mathbb Z^k$-actions generated by finitely many commuting homeomorphisms. Notably, no amenability assumption is needed: we explain why spectral-gap phenomena for non-amenable actions, which do produce cohomological stability in Hölder, Sobolev and $L^2$ categories, are invisible to the uniform norm. We also discuss the genuinely different smooth category and state a conjecture regarding cohomological $C^\infty$-stability of $\mathbb Z^k$-action by smooth circle diffeomorphisms without periodic orbits, connecting the problem with works of Moser, Fayad-Khanin, Avila-Kocsard and Petković.

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arXiv2026

Hardy-Rogers and Jungck Type Fixed Point Theorems in Perturbed Metric Spaces, Stability and Data Dependence

In this paper we establish Hardy-Rogers and Jungck type fixed point theorems in perturbed metric spaces, where the observed distance $D$ is separated from the exact metric $d$ by a nonnegative perturbation $P$. Rather than the uniform absorption of $P$ by $d$ required under domination, we examine a weaker demand, imposed only on the pairs of points that appear with a contractive coefficient. We show that without some such condition, and without a continuity hypothesis on $T$, a perturbed Banach contraction on a complete perturbed metric space may fail to have a fixed point. We further prove Ulam-Hyers stability, well-posedness, and data dependence results in which residuals are measured in the observed distance $D$, with all constants explicit, and we derive a priori error estimates for the Picard and Jungck iterations computable from observed data. The Jungck type theorem is established for weakly compatible pairs.

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arXiv2022

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+ω(y), y)$ and $T_α(x, y)=(x+α, y)$, where $ω(y)$ is non-degenerate and $α$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^\infty$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $\mathbb{Z}^2$-actions on the cylinder, generated by commuting twist maps.

math.DS↗

arXiv2018

Local Rigidity for Simultaneous Diophantine Translations on Tori of Arbitrary Dimension

We show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This generalizes recent result of Karaliolios [4] of the action generators to higher rank abelian actions, and the result of Moser [8] to higher dimensional tori.

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