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Piotr Oprocha

Publications and source records attributed to Piotr Oprocha.

At least 19 recordsLinked to original sources
arXiv2026

Shadowing in the presence of singularities: oriented versus standard shadowing, entropy and the structure of recurrent sets

We study two shadowing properties for flows that differ in the allowed reparametrizations of time: oriented shadowing permits arbitrary increasing reparametrizations, whereas standard shadowing requires their distortion to be uniformly close to one. We prove that these notions are distinct already for $C^\infty$ flows on every closed oriented surface. Moreover, such examples are $C^0$-dense among $C^1$ flows with a singularity and consequently, on closed oriented surfaces with non-zero Euler characteristic, they are dense among all $C^1$ flows. We then relate local standard shadowing to recurrence and entropy. A non-trivial chain-transitive set with local standard shadowing forces positive topological entropy unless it is an irreducible almost heteroclinic set. Consequently, for a zero-entropy flow with standard shadowing, every non-trivial chain-recurrent class has this form, and every non-singular one is minimal. For surface flows, we further prove that oriented shadowing together with finitely many singularities forces every chain-recurrent class to be minimal.

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arXiv2026

Special $\alpha$-limit sets in the hyperspace of continua: closedness and entropy for interval maps

This paper investigates the backward dynamics of hyperspace systems induced by continuous interval maps. Focusing on the hyperspace of continua, we provide a structural characterization of the $\alpha$-limit sets of backward branches for interval subcontinua. A central result of this work is the proof that the special $\alpha$-limit set of any nondegenerate subinterval is always closed. This reveals a striking topological contrast with classical single-point dynamics, where special $\alpha$-limit sets need not be closed. Finally, we prove that if a special $\alpha$-limit set contains two nondegenerate periodic continua with disjoint orbits, then the base map has a horseshoe and, consequently, positive topological entropy.

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arXiv2026

Measures of maximal entropy for Markovian dynamics on the Gehman dendrite

We study transitive dynamical systems on the Gehman dendrite $\mathcal{G}$ for which the endpoint set $\mathrm{End}(\mathcal{G})$ is invariant. Our goal is to approximate such systems by maps whose measure-theoretic behaviour at maximal entropy is governed by an explicit countable Markov structure. We introduce a class of Markovian maps, encode their dynamics by countable Markov graphs, and use the criteria of Vere-Jones, Gurevich, Salama and Ruette to control the existence of measures of maximal entropy. The main theorem gives two arbitrarily close mixing Markovian perturbations of any given system in the considered class: one has a unique measure of maximal entropy, while the other has none.

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arXiv2026

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities

We establish a complete structural classification for continuum-wise hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw$_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus $\mathbb{T}^2$ or to its standard hyperelliptic quotient on the sphere $\mathbb{S}^2$. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.

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arXiv2026

On limit sets and equicontinuity in the hyperspace of continua in dimension one

The paper studies the structure of $\omega$-limit sets of map $\tilde{f}$ induced on the hyperspace $C(G)$ of all connected compact sets, by dynamical system $(G,f)$ acting on a topological graph $G$. In the case of the base space being a topological tree we additionally show that $\tilde{f}$ is always almost equicontinuous and characterize its Birkhoff center.

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arXiv2026

On Lebesgue measure preserving Besicovitch functions

We consider the space $C_{\lambda}$ of all continuous interval maps preserving the Lebesgue measure $\lambda$. A continuous function $f\colon~[0,1]\to \mathbb R$ is called Besicovitch if it does not have any finite or infinite unilateral derivative. It is known that the set of Besicovitch functions in $C_{\lambda}$ is nonempty and meager. We prove that no Besicovitch function is invertible $\lambda$-almost everywhere. As a consequence, every Besicovitch function in $C_{\lambda}$ has positive measure-theoretic entropy with respect to $\lambda$. Furthermore, we show that Besicovitch functions are dense in $C_{\lambda}$ and, consequently, also dense in the class of interval maps with a dense set of periodic points.

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arXiv2025

Observable Dynamics and the Generic Coincidence of Milnor, Statistical, and Physical Attractors

We study observable dynamics of generic continuous interval maps. We prove that, for a residual subset of $C([0,1])$, the global Milnor, statistical, and physical attractors coincide with the non-wandering set. Thus, for a typical interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics. The common attractor is a robust Cantor set of zero Hausdorff dimension. Moreover, every invariant probability measure has a basin of zero Lebesgue measure. We also present examples showing that, outside the generic setting, positive topological entropy may occur outside the global Milnor attractor, and the three attractors need not coincide.

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arXiv2025

Spectrum of invariant measures via generic points

We describe the spectrum of an ergodic invariant measure by examining the behaviour of its generic points. We define regular Wiener--Wintner generic points for a measure to generalise the characterisation of generic points for discrete spectrum measure from Lenz et al. [Ergodic Theory and Dynamical Systems vol. \textbf{44} (2024), no. 2, 524--568]. We also study limits of sequences of generic points with respect to the Besicovitch pseudometric. This translates to results about limits of measures with respect to the metric rho-bar $\bar{\rho}$ generalising Ornstein's d-bar metric. We study how the spectrum behaves when passing to the limit and we prove that points generic for discrete spectrum, totally ergodic, or (weakly) mixing measures, property K, zero entropy measures form a closed set with respect to the Besicovitch pseudometric. Hence, the same holds for corresponding measures with respect to the rho-bar metric. Our methods have already been used to prove existence of ergodic measures with desired properties, in particular with discrete spectrum. They also lead to a new proof of rational discrete spectrum of the Mirsky measure associated with a given set of $\mathscr B$-free numbers.

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arXiv2025

On the cohomology of homshifts

We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour $\mathbb{Z}^d$ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no.1, 237-269) we give a necessary and sufficient condition for homshifts to be cohomological trivial. This condition is expressed in terms of the topology of a natural two-dimensional CW complex arising from the shift space which can be analyzed in many natural cases. However, we prove that in general, cohomological triviality is algorithmically undecidable for homshifts.

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arXiv2025

Undecidability of the block gluing classes of homshifts

A homshift is a $d$-dimensional shift of finite type which arises as the space of graph homomorphisms from the grid graph $\mathbb Z^d$ to a finite connected undirected graph $G$. While shifts of finite type are known to be mired by the swamp of undecidability, homshifts seem to be better behaved and there was hope that all the properties of homshifts are decidable. In this paper we build on the work by Gangloff, Hellouin de Menibus and Oprocha (arxiv:2211.04075) to show that finer mixing properties are undecidable for reasons completely different than the ones used to prove undecidability for general multidimensional shifts of finite type. Inspired by the work of Gao, Jackson, Krohne and Seward (arxiv:1803.03872) and elementary algebraic topology, we interpret the square cover introduced by Gangloff, Hellouin de Menibus and Oprocha topologically. Using this interpretation, we prove that it is undecidable whether a homshift is $\Theta(n)$-block gluing or not, by relating this problem to the one of finiteness for finitely presented groups.

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arXiv2025

Entropy Flexibility of Dynamical Systems

Inspired by Katok's intermediate entropy property [Inst. Hautes \'Etudes Sci. Publ. Math. 51 (1980), 137-173], we introduce and study the notion of entropy flexibility for discrete-time and continuous-time dynamical systems. By using renewal systems techniques, we show that this property is present in several classes of systems where any intermediate value of entropy can be attained on a strictly ergodic sub-system. In addition, we prove an entropy flexibility analogue of Katok's conjecture: Entropy flexibility is a typical property for vector fields on 3-manifolds and surface diffeomorphisms.

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arXiv2025

On entropy of pure mixing maps on dendrites

For every $0<\alpha\le\infty$ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy $\alpha$. It has been previously shown by \v{S}pitalsk\'y that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.

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arXiv2025

Continuum-wise hyperbolicity, periodic shadowing, and measures of maximal entropy

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of $\mathbb{S}^2$, and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpi\'nski Carpet defined in [12], which does not satisfy the specification property.

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arXiv2025

Local Shadowing Beyond Global Shadowing: Entropy and Dense Manifold Realizations

Shadowable points were developed to cover cases in which a local shadowing mechanism survives without a global shadowing property. We show that on every compact manifold of dimension at least two, there is a $C^0$-dense set $\mathcal{R}$ of homeomorphisms so that each $f\in \mathcal{R}$ has a transitive chain component $D$ consisting of shadowable points, although $f$, $f|_D$, and every chain recurrent class meeting a neighborhood of $D$ fail shadowing. Thus ambient pointwise tracing is neither inherited from global shadowing nor explained by a shadowing subsystem. Furthermore, on general compact spaces, arbitrary Cantor dynamics can occur as the entire set of shadowable points. We also study shadowable points through the local dynamics of chain classes and derive, under additional hypotheses, semi-horseshoes, entropy-bearing ergodic approximations of measures, and entropy flexibility.

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arXiv2025

Rigidity and Toeplitz systems

The aim of this paper is to study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems and enumeration systems. We use Bratteli diagrams to control invariant measures that are produced in our constructions. This leads to systems with desired properties. Among other things, we show that there exist Toeplitz systems with zero entropy which are not partially measure-theoretically rigid with respect to any of its invariant measures. We investigate enumeration systems defined by a linear recursion, prove that all such systems are partially rigid and present an example of an enumeration system which is not measure-theoretically rigid. We construct a minimal $\mathcal{S}$-adic Toeplitz subshift which has countably infinitely many ergodic invariant probability measures which are rigid for the same rigidity sequence.

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arXiv2025

On recurrence and entropy in hyperspace of continua in dimension one

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$. This is well known that it does not hold on the larger hyperspace of all compact subsets. Also negative examples were given for the hyperspace $C(X)$ on some continua $X$, including dendrites. Our work extends previous positive results obtained first for much simpler case of compact interval by completely different tools.

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arXiv2024

Cantor subsystems on the Gehman dendrite

In the present note we focus on dynamics on the Gehman dendrite $\mathcal{G}$. It is well-known that the set of its endpoints is homeomorphic to a standard Cantor ternary set. For any given surjective Cantor system $\mathcal{C}$ we provide constructions of (i) a mixing but not exact and (ii) an exact map on $\mathcal{G}$, such that in both cases the subsystem formed by $\text{End}(\mathcal{G})$ is conjugate to the initially chosen system on $\mathcal{C}$.

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