Polynomial entropy of induced homeomorphisms on $C_k(\mathbb{S}^1)$ and $C_k([0,1])$
We compute the polynomial entropy of $C_k(f)$ where $f$ is any circle or interval homeomorphism.
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arXiv subjects
Jelena Katić
Publications and source records attributed to Jelena Katić.
A flexibility result for polynomial entropy of pointwise periodic homeomorphisms
We construct a family of continua and pointwise periodic homeomorphisms realizing arbitrary polynomial entropy values in $[0,+\infty]$. In particular, this provides examples of pointwise periodic homeomorphisms with positive polynomial entropy. This contrasts with the fact that pointwise periodic homeomorphisms on connected manifolds and local dendrites have zero polynomial entropy.
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Complexity function and entropy of induced maps on hyperspaces of continua
We use the complexity function of an invariant, not necessary closed, subset of a two-sided shift space to compute the polynomial entropy of the induced dynamics on the hyperspace of continua for certain one-dimensional dynamical systems. We also provide a simple criterion for $f$ that implies $C(f)$ has infinite topological entropy.
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Polynomial entropy on regular curves
We show that the polynomial entropy of homeomorphisms on regular curves is bounded above by one. Moreover, the polynomial entropy equals one under the fairly mild condition that the homeomorphism possesses a wandering point. We obtain a rigidity result for homeomorphisms on local dendrites (and therefore on graphs and dendrites): their polynomial entropy is either zero or one.
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Shadowing property on hyperspace of continua induced by Morse gradient system
It is known that Morse-Smale diffeomorphisms have the shadowing property; however, the question of whether $C(f)$ also has the shadowing property when $f$ is Morse-Smale remains open and has been resolved only in a few specific cases~\cite{AB}. We prove that if $f:M\to M$ is a time-one-map of Morse gradient flow, the induced map $C(f):C(M)\to C(M)$ on the hyperspace of subcontinua does not have the shadowing property.
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Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory
We compare spectral invariants in periodical orbits and Lagrangian Floer homology case, for closed symplectic manifold $P$ and its closed Lagrangian submanifolds $L$, when $ω|_{π_2(P,L)}=0$, and $μ|_{π_2(P,L)}=0$. From this result, we derive a corollary considering comparison of Hofer's distance in periodic orbits and Lagrangian case. We also define a product $HF_*(H)\otimes HF_*(L,ϕ^1_H(L))\to HF_*(L,ϕ^1_H(L))$ and prove subadditivity of invariants with respect to this product.
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Polynomial entropy of induced maps of circle and interval homeomorphisms
We compute the polynomial entropy of the induced maps on hyperspace for a homeomorphism $f$ of an interval or a circle with finitely many non-wandering points.
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A note on partial quasi-morphisms and products in Lagrangian Floer homology in cotangent bundles
We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.
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Spectral numbers and manifolds with boundary
We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(α,H)$ to every singular homological class $α\in H_*(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^*M$. We also derive certain properties of spectral numbers, for example we prove that spectral invariants $c_\pm(H,N)$ associated to the whole Floer homology $HF_*(H,N:M)$ of the submanifold $N$, are limits of the decreasing nested family of open sets.
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Spectral Invariants in Lagrangian Floer homology of open subset
We define and investigate spectral invariants for Floer homology $HF(H,U:M)$ of an open subset $U\subset M$ in $T^*M$, defined by Kasturirangan and Oh as a direct limit of Floer homologies of approximations. We define a module structure product on $HF(H,U:M)$ and prove the triangle inequality for invariants with respect to this product. We also prove the continuity of these invariants and compare them with spectral invariants for periodic orbits case in $T^*M$.
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