Mathematics and Statistics - Publications

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    Chaotic extensions of continuous maps on compact manifolds
    ( 2017-09-02) Moothathu, T. K.Subrahmonian
    Let X be a compact connected Lipschitz manifold, with or without boundary. We show that for every continuous map f : X → X and every nowhere dense closed subset K of X, there is a topologically transitive continuous map g : X → X having a dense set of periodic points in X such that g|K = f |K. Combined with a deep theorem of Sullivan, our result extends to all compact connected topological manifolds of dimension ≠.
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    Midsets and Voronoi type decomposition with respect to closed convex sets
    ( 2018-01-01) Subrahmonian Moothathu, T. K.
    Let Ωk denote the collection of all nonempty closed convex subsets of ℝk. We provide short proofs for the following: (i) {x ∈ ℝk: dist(x, A) = ϵ} is a C1-manifold of dimension k-1 for every A ∈ Ωk \ {ℝk} and ϵ > 0, (ii) {x ∈ ℝk: dist(x, A) = dist(x, B)} is a C1-manifold of dimension k - 1 for any two disjoint A, B Ωk.We also study the distance of points in ℝk to finitely many closed convex sets. Let k, n ≤ 2 and A = ∪j=1n Aj, where A1, ⋯, An ∈ Ωk are pairwise disjoint. We consider a Voronoi type decomposition of ℝk and establish some topological properties of its 'conflict set'. Letting Xp = {x ∈ ℝk: |{a ∈ A: ||x - a|| = dist(x,A)}| = p}, we prove with the help of result (ii) stated above that X1 ∪ X2 is a connected dense open subset of ℝk and that X2 = ∪p=2n Xp.
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    Lineability in the sets of Baire and continuous real functions
    ( 2018-02-15) Moothathu, T. K.Subrahmonian
    Discovering large algebraic structures within various sets of exotic real-valued (or complex-valued) functions or sequences is a very active research program now. For a cardinal number η a subset A of a vector space X is said to be η-lineable if there is a vector subspace Z⊂X of dimension η with Z⊂A∪{0}. In this article, we investigate lineability properties in certain vector spaces of Baire-two functions and continuous real-valued functions.
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    Corrigendum to “Syndetically proximal pairs” [J. Math. Anal. Appl. 379 (2011) 656–663](S0022247X11000874)(10.1016/j.jmaa.2011.01.060)
    ( 2018-03-01) Li, Jian ; Moothathu, T. K.Subrahmonian ; Oprocha, Piotr
    We give a counterexample to Theorem 9 in “Syndetically proximal pairs” [J. Math. Anal. Appl. 379 (2011) 656–663]. We also provide sufficient conditions for the conclusion of Theorem 9 to hold.
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    Linear independence of a hypercyclic orbit for semigroups
    ( 2018-11-01) Moothathu, T. K.Subrahmonian
    A question of Conejero and Peris from 2005 asks the following: if {T(t)}t≥0 is a strongly continuous semigroup of operators acting on a Banach space, is the orbit {T(t)x:t≥0} of a hypercyclic vector x always linearly independent? We give a positive answer to this question for complex Banach spaces under the assumption that the spectrum σ(A) of the generator A of the semigroup is nonempty.