Let T)(x) cos(,arccosx), 1 _< x _< 1, where , > 1 is not an integer. For a certain set o... more Let T)(x) cos(,arccosx), 1 _< x _< 1, where , > 1 is not an integer. For a certain set of ,’s which are irrational, the density of the unique absolutely continuous measure invariant under T,X is determined exactly. This is accomplished by showing that T,X is differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed as the unique left eigenvector of a matrix.
We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, whic... more We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..
Abstract: A Markov switching position dependent random map is a random map of a finite number of ... more Abstract: A Markov switching position dependent random map is a random map of a finite number of measurable transformations where the probability of switching from one transformation to another is controlled by a position dependent irreducible stochastic matrix W. Existence of ...
We present a definition of time measurement based on high energy photons and the fundamental leng... more We present a definition of time measurement based on high energy photons and the fundamental length scale, and show that, for macroscopic time, it is in accord with the Lorentz transformation of special relativity. To do this we define observer in a different way than in special relativity.
Proceedings of the American Mathematical Society, 2013
For a large class of piecewise expanding C 1 , 1 \mathcal {C}^{1,1} maps of the interval we prove... more For a large class of piecewise expanding C 1 , 1 \mathcal {C}^{1,1} maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known 2 / inf | τ ′ | 2/\inf |\tau ’| . Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim’s for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps.
Let T)(x) cos(,arccosx), 1 _< x _< 1, where , > 1 is not an integer. For a certain set o... more Let T)(x) cos(,arccosx), 1 _< x _< 1, where , > 1 is not an integer. For a certain set of ,’s which are irrational, the density of the unique absolutely continuous measure invariant under T,X is determined exactly. This is accomplished by showing that T,X is differentially conjugate to a piecewise linear Markov map whose unique invariant density can be computed as the unique left eigenvector of a matrix.
We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, whic... more We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..
Abstract: A Markov switching position dependent random map is a random map of a finite number of ... more Abstract: A Markov switching position dependent random map is a random map of a finite number of measurable transformations where the probability of switching from one transformation to another is controlled by a position dependent irreducible stochastic matrix W. Existence of ...
We present a definition of time measurement based on high energy photons and the fundamental leng... more We present a definition of time measurement based on high energy photons and the fundamental length scale, and show that, for macroscopic time, it is in accord with the Lorentz transformation of special relativity. To do this we define observer in a different way than in special relativity.
Proceedings of the American Mathematical Society, 2013
For a large class of piecewise expanding C 1 , 1 \mathcal {C}^{1,1} maps of the interval we prove... more For a large class of piecewise expanding C 1 , 1 \mathcal {C}^{1,1} maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known 2 / inf | τ ′ | 2/\inf |\tau ’| . Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim’s for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps.
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