arXiv subjects

Young Woo Nam

Publications and source records attributed to Young Woo Nam.

11 recordsLinked to original sources

arXiv2024

Hyers-Ulam stability of the first order difference equation with average growth rate

The first order difference equation induced by the sequence of maps on $ \mathbb{C} $ has Hyers-Ulam stability where the limit of the geometric average of growth rate is convergent and not equal to one. %The average growth rate is a generalization of contracting or expanding constant of maps. We show no Hyers-Ulam stability where the average growth rate is (pre)periodic even though each periodic growth rate is strictly less than one. Examples of difference equation generated by time dependent maps which contains contracting maps and expanding maps are given.

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arXiv2022

Hyers-Ulam stability of the first order difference equation generated by linear maps

Hyers-Ulam stability of the difference equation $ z_{n+1} = a_nz_n + b_n $ is investigated. If $ \prod_{j=1}^{n}|a_j| $ has subexponential growth rate, then difference equation generated by linear maps has no Hyers-Ulam stability. Other complementary results are also found where $ \lim_{n \rightarrow \infty} \left(\prod_{j=1}^{n}|a_j| \right)^{\frac{1}{n}} $ is greater or less than one. These results contain Hyers-Ulam stability of the first order linear difference equation with periodic coefficients also.

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arXiv2019

Hyers-Ulam stability of parabolic Möbius difference equation

The linear fractional map $ g(z) = \frac{az+ b}{cz + d} $ on the Riemann sphere with complex coefficients $ ad-bc = 1 $ is and $ a+d = \pm 2 $, then $ g $ is called {\em parabolic} Möbius map. Let $ \{ b_n \}_{n \in \mathbb{N}_0} $ be the solution of the parabolic Möbius difference equation $ b_{n+1} = g(b_n) $ for every $ n \in \mathbb{N}_0 $. We show that the sequence $ \{ b_n \}_{n \in \mathbb{N}_0} $ has no Hyers-Ulam stability.

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arXiv2018

Hyers-Ulam stability of loxodromic Möbius difference equation

Hyers-Ulam of the sequence $ \{z_n\}_{n \in \mathbb{N}} $ satisfying the difference equation $ z_{i+1} = g(z_i) $ where $ g(z) = \frac{az + b}{cz + d} $ with complex numbers $ a $, $ b $, $ c $ and $ d $ is defined. Let $ g $ be loxodromic Möbius map, that is, $ g $ satisfies that $ ad-bc =1 $ and $a + d \in \mathbb{C} \setminus [-2,2] $. Hyers-Ulam stability holds if the initial point of $ \{z_n\}_{n \in \mathbb{N}} $ is in the exterior of avoided region, which is the union of the certain disks of $ g^{-n}(\infty) $ for all $ n \in \mathbb{N} $.

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arXiv2017

Hyers-Ulam stability of hyperbolic Möbius difference equation

Hyers-Ulam stability of the difference equation with the initial point $ z_0 $ as follows $$ z_{i+1} = \frac{az_i + b}{cz_i + d} $$ is investigated for complex numbers $ a,b,c $ and $ d $ where $ ad - bc = 1 $, $ c \neq 0 $ and $a + d \in \mathbb{R} \setminus [-2,2] $. The stability of the sequence $ \{z_n\}_{n \in \mathbb{N}_0} $ holds if the initial point is in the exterior of a certain disk of which center is $ -\frac{d}{c} $. Furthermore, the region for stability can be extended to the complement of some neighborhood of the line segment between $ -\frac{d}{c} $ and the repelling fixed point of the map $ z \mapsto \frac{az + b}{cz + d} $. This result is the generalization of Hyers-Ulam stability of Pielou logistic equation.

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arXiv2017

Hyers-Ulam stability of elliptic Möbius difference equation

The linear fractional map $ f(z) = \frac{az+ b}{cz + d} $ on the Riemann sphere with complex coefficients $ ad-bc \neq 0 $ is called Möbius map. If $ f $ satisfies $ ad-bc=1 $ and $ -2<a+d<2 $, then $ f $ is called $\textit{elliptic}$ Möbius map. Let $ \{ b_n \}_{n \in \mathbb{N}_0} $ be the solution of the elliptic Möbius difference equation $ b_{n+1} = f(b_n) $ for every $ n \in \mathbb{N}_0 $. Then the sequence $ \{ b_n \}_{n \in \mathbb{N}_0} $ has no Hyers-Ulam stability.

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arXiv2015

Hénon renormalization in arbitrary dimension : Invariant space under renormalization operator

Infinitely renormalizable Hénon-like map in arbitrary finite dimension is considered. The set, $\mathcal N$ of infinitely renormalizable Hénon-like maps satisfying the certain condition is invariant under renormalization operator. The Cantor attractor of infinitely renormalizable Hénon-like map, $F$ in $\mathcal N$ has {\em unbounded geometry} almost everywhere in the parameter space of the universal number which corresponds to the average Jacobian of two dimensional map. This is an extension of the same result in $\mathcal N$ for three dimensional infinitely renormalizable Hénon-like maps.

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arXiv2015

Invariant space under Hénon renormalization : Intrinsic geometry of Cantor attractor

Three dimensional Hńon-like map $$ F(x,y,z) = (f(x) - ε(x,y,z),\ x,\ δ(x,y,z)) $$ is defined on the cubic box $ B $. An invariant space under renormalization would appear only in higher dimension. Consider renormalizable maps each of which satisfies the condition $$ \partial_y δ\circ F(x,y,z) + \partial_z δ\circ F(x,y,z) \cdot \partial_x δ(x,y,z) \equiv 0 $$ for $ (x,y,z) \in B $. Denote the set of maps satisfying the above condition be $ \mathcal N $. Then the set $ \mathcal N \cap \mathcal I(\bar ε) $ is invariant under the renormalization operator where $ \mathcal I(\bar ε) $ is the set of infinitely renormalizable maps. Hénon like diffeomorphism in $ \mathcal N \cap \mathcal I(\bar ε) $ has universal numbers, $ b_2 \asymp | \partial_z δ| $ and $ b_1 = b_F /b_2 $ where $ b_F $ is the average Jacobian of $ F $. The Cantor attractor of $ F \in \mathcal N \cap \mathcal I(\bar ε) $, $ \mathcal O_F $ has {\em unbounded geometry} almost everywhere in the parameter space of $ b_1 $. If two maps in $ \mathcal N $ has different universal numbers $ b_1 $ and $ \widetilde b_1 $, then the homeomorphism between two Cantor attractor is at most Hölder continuous, which is called {\em non rigidity}.

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arXiv2015

Renormalization of Hénon map in arbitrary dimension I : Universality and reduction of ambient space

Period doubling Hénon renormalization of strongly dissipative maps is generalized in arbitrary finite dimension. In particular, a small perturbation of toy model maps with dominated splitting has invariant $C^r$ surfaces embedded in higher dimension and the Cantor attractor has unbounded geometry with respect to full Lebesgue measure on the parameter space. It is an extension of dynamical properties of three dimensional infinitely renormalizable Hénon-like map in arbitrary finite dimension.

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arXiv2014

Renormalization of $C^r$ Hénon map : Two dimensional embedded map in three dimension

We study renormalization of highly dissipative analytic three dimensional Hénon maps $$ F(x,y,z) = (f(x) - \varepsilon(x,y,z),\ x,\ δ(x,y,z)) $$ where $ \varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \varepsilon_{2d}(x,y) $. Under certain conditions, $ C^r $ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $ 2 \leq r < \infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ Hénon-like map. It also defines two dimensional embedded $ C^r $ Hénon-like maps in three dimension. In this class, universality theorem is re-constructed by conjugation. Geometric properties on the critical Cantor set in invariant surfaces are the same as those of two dimensional maps --- non existence of the continuous line field and unbounded geometry. The set of embedded two dimensional Hénon-like maps is open and dense subset of the parameter space of average Jacobian, $ b_{F_{2d}} $ for any given smoothness, $ 2 \leq r < \infty $.

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arXiv2014

Renormalization of three dimensional Hénon map I : Reduction of ambient space

Three dimensional analytic Hénon-like map $$ F(x,y,z) = (f(x) - ε(x,y,z),\, x,\, δ(x,y,z)) $$ and its {\em period doubling} renormalization is defined. If $ F $ is infinitely renormalizable map, Jacobian determinant of $ n^{th} $ renormalized map, $ R^nF $ has asymptotically universal expression $$ Jac R^nF = b_F^{2^n}a(x)(1 + O(ρ^n)) $$ where $ b_F $ is the average Jacobian of $ F $. The toy model map, $ F_{mod} $ is defined as the map satisfying $ \partial_z ε\equiv 0 $. The set of toy model map is invariant under renormalizaton. Moreover, if $ \| \partial_z δ\| \ll \| \partial_y ε\| $, then there exists the continuous invariant plane field over $ \mathcal O_F $ with dominated splitting. Under this condition, three dimensional Hénon-like map %with the dominated splitting is dynamically decomposed into two dimensional map with contraction along the strong stable direction. Any invariant line field on this plane filed over $ \mathcal O_{F_{mod}} $ cannot be continuous.

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