Invariant densities for composed piecewise fractional linear maps

AbstractWe find an explicit formula for the invariant densityhof an arbitrary eventually expanding piecewise linear mapτof an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrixSwhich is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue ofS, then the dynamical system (τ,h⋅m) is ergodic with full support.

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Family of piecewise expanding maps having singular measure as a limit of ACIMs

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385711000836 ◽

2011 ◽

Vol 33 (1) ◽

pp. 158-167 ◽

Cited By ~ 5

Author(s):

ZHENYANG LI ◽

PAWEŁ GÓRA ◽

ABRAHAM BOYARSKY ◽

HARALD PROPPE ◽

PEYMAN ESLAMI

Keyword(s):

Piecewise Linear ◽

Dynamical Behavior ◽

Singular Measure ◽

Expanding Maps ◽

Linear Maps ◽

Absolutely Continuous Invariant Measure ◽

Piecewise Linear Maps ◽

Point Measure ◽

Invariant Densities ◽

Absolutely Continuous Invariant

AbstractKeller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

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Linear maps preserving structured singular values of matrices

Linear Algebra and its Applications ◽

10.1016/j.laa.2021.03.002 ◽

2021 ◽

Author(s):

Constantin Costara

Keyword(s):

Singular Values ◽

Linear Maps

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Further refinements of Young’s type inequality for positive linear maps

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01032-4 ◽

2021 ◽

Vol 115 (2) ◽

Author(s):

Mohamed Amine Ighachane ◽

Mohamed Akkouchi

Keyword(s):

Type Inequality ◽

Linear Maps ◽

Positive Linear Maps

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An Algebraic Brascamp–Lieb Inequality

Journal of Geometric Analysis ◽

10.1007/s12220-021-00638-9 ◽

2021 ◽

Author(s):

Jennifer Duncan

Keyword(s):

General Class ◽

Recent Progress ◽

Algebraic Groups ◽

Hölder’S Inequality ◽

Duke Math ◽

Rational Maps ◽

Affine Invariant ◽

Linear Maps ◽

Nonlinear Maps ◽

Funct Anal

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

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How Many Inflections are There in the Lyapunov Spectrum?

Communications in Mathematical Physics ◽

10.1007/s00220-021-04161-4 ◽

2021 ◽

Author(s):

O. Jenkinson ◽

M. Pollicott ◽

P. Vytnova

Keyword(s):

Piecewise Linear ◽

Lyapunov Spectrum ◽

Stat Phys ◽

Linear Map ◽

Piecewise Linear Map ◽

Expanding Map ◽

Lyapunov Spectra ◽

Linear Maps ◽

Inflection Points ◽

Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

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