On Graph-realizable Sets of Periods

Qualitative Theory of Dynamical Systems, 2015

In this paper we characterize all possible sets of periods of homeomorphisms defined on some classes of finite connected compact graphs.

Proceedings of the American Mathematical Society

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

Dynamics Reported, 1995

We construct the "spectral" decomposition of the sets P er f , ω(f ) = ∪ω (x) and Ω(f ) for a continuous map f : [0, 1] → [0, 1]. Several corollaries are obtained; the main ones describe the generic properties of f -invariant measures, the structure of the set Ω(f ) \ P er f and the generic limit behavior of an orbit for maps without wandering intervals. The "spectral" decomposition for piecewise-monotone maps is deduced from the Decomposition Theorem. Finally we explain how to extend the results of the present paper for a continuous map of a one-dimensional branched manifold into itself.

Rocky Mountain Journal of Mathematics, 2017

The objective of the present work is to present information on the set of periodic points of a continuous self-map on a closed surface which can be obtained using the action of this map on homological groups of the closed surface.

Discrete Mathematics, 2009

We show that line graphs G = L(H) with σ 2 (G) ≥ 7 contain cycles of all lengths k, 2 rad(H) + 1 ≤ k ≤ c(G). This implies that every line graph of such a graph with 2 rad(H) ≥ ∆(H) is subpancyclic, improving a recent result of Xiong and Li. The bound on σ 2 (G) is best possible.

arXiv: Spectral Theory, 2018

We consider Laplacians on periodic discrete graphs. The spectrum of the Laplacian consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain the following results: (1) a localization of each band in terms of eigenvalues of Laplacians on some auxiliary finite graphs; (2) estimates of the Lebesgue measure of the Laplacian spectrum in terms of geometric invariants for periodic graphs (we show that these estimates become identities for specific graphs); (3) a special decomposition of the Laplacian into a direct integral, where fiber Laplacians have the minimal number of coefficients depending on the quasimomentum; (4) necessary and sufficient conditions for matrices on a finite graph to be fiber Laplacians. Moreover, similar results for Schr\"odinger operators with periodic potentials are obtained.

Communications in Mathematical Physics, 2006

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z n . It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only).

International Mathematics Research Notices

We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree, which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non-quasi-isometric Cayley graphs with the same Laplacian spectrum, a union of two intervals, which we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another. For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct, moreover, a c...

Springer Proceedings in Mathematics & Statistics, 2016

The goal of this paper is to show what information on the set of periodic points of a homeomorphism on a closed surface can be obtained using the action of this homeomorphism on the homological groups of the closed surface.

Electronic Journal of Graph Theory and Applications

For a finite connected graph X, we consider the graph RX obtained from X by associating a new vertex to every edge of X and joining by edges the extremities of each edge of X to the corresponding new vertex. We express the spectrum of the Laplace operator on RX as a function of the corresponding spectrum on X. As a corollary, we show that X is a complete graph if and only if λ 1 (RX) > 1 2 . We give a re-interpretation of the correspondence X → RX in terms of the right-angled Coxeter group defined by X.