
Previous Article
Global exponential stability of traveling waves in monotone bistable systems
 DCDS Home
 This Issue

Next Article
Almost sure stability of some stochastic dynamical systems with memory
Infinite Bernoulli convolutions with different probabilities
1.  Institute of Mathematics, Budapest University of Technology and Economics, P.O.box 91, H1529, Budapest, Hungary 
$Y_{\lambda}^p$:=$\sum_{n=0}^{\infty}\pm \lambda ^n$
where the "$+$" and "$$" signs are chosen independently with
probability $p$ and $1p$. Let $\nu_\lambda^p$ be the distribution
of the random sum $\nu_\lambda^p(E)$:=$Prob(Y_{\lambda}^p \in
E)$ for every set $E$. The conjecture is that for every $p \in
(0,1)$ the measure $\nu_\lambda^p$ is absolutely continuous w.r.t.
Lebesgue measure and with the density in $L^2(R)$ for almost every
$\lambda\in (p^p\cdot(1p)^{(1p)},1).$
B. Solomyak and Y. Peres [3, [Corollary 1.4] proved that
for every $p \in (\frac{1}{3},\frac{2}{3})$ the distribution
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda \in (p^2+(1p)^2,1).$ In this paper we
extend the parameter interval where a weakened version of the
conjecture still holds. Namely, we prove Corollary 3 that
for every $p \in (0,\frac{1}{3}]$ the measure
$\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for
almost every $\lambda\in(F(p),1)$, where
$F(p)=(12p)^{2\log41/\log 9}$, see Figure 3.
[1] 
Christoph Bandt, Helena PeÑa. Polynomial approximation of selfsimilar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 46114623. doi: 10.3934/dcds.2017198 
[2] 
Qingsong Gu, Jiaxin Hu, SzeMan Ngai. Geometry of selfsimilar measures on intervals with overlaps and applications to subGaussian heat kernel estimates. Communications on Pure & Applied Analysis, 2020, 19 (2) : 641676. doi: 10.3934/cpaa.2020030 
[3] 
Weronika Biedrzycka, Marta TyranKamińska. Selfsimilar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems  B, 2018, 23 (1) : 1327. doi: 10.3934/dcdsb.2018002 
[4] 
Marco Cannone, Grzegorz Karch. On selfsimilar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801808. doi: 10.3934/krm.2013.6.801 
[5] 
Rostislav Grigorchuk, Volodymyr Nekrashevych. Selfsimilar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323370. doi: 10.3934/jmd.2007.1.323 
[6] 
Anna Chiara Lai, Paola Loreti. Selfsimilar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401419. doi: 10.3934/nhm.2015.10.401 
[7] 
Kin Ming Hui. Existence of selfsimilar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 863880. doi: 10.3934/dcds.2019036 
[8] 
D. G. Aronson. Selfsimilar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems  B, 2012, 17 (6) : 16851691. doi: 10.3934/dcdsb.2012.17.1685 
[9] 
G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically selfsimilar dynamics. Conference Publications, 2005, 2005 (Special) : 131141. doi: 10.3934/proc.2005.2005.131 
[10] 
Qiaolin He. Numerical simulation and selfsimilar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems  B, 2010, 13 (1) : 101116. doi: 10.3934/dcdsb.2010.13.101 
[11] 
Bendong Lou. Selfsimilar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857879. doi: 10.3934/nhm.2012.7.857 
[12] 
F. Berezovskaya, G. Karev. Bifurcations of selfsimilar solutions of the FokkerPlank equations. Conference Publications, 2005, 2005 (Special) : 9199. doi: 10.3934/proc.2005.2005.91 
[13] 
Shota Sato, Eiji Yanagida. Singular backward selfsimilar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems  S, 2011, 4 (4) : 897906. doi: 10.3934/dcdss.2011.4.897 
[14] 
Alberto Bressan, Wen Shen. A posteriori error estimates for selfsimilar solutions to the Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 113130. doi: 10.3934/dcds.2020168 
[15] 
Shota Sato, Eiji Yanagida. Forward selfsimilar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313331. doi: 10.3934/dcds.2010.26.313 
[16] 
L. Olsen. Rates of convergence towards the boundary of a selfsimilar set. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 799811. doi: 10.3934/dcds.2007.19.799 
[17] 
Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to selfsimilar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 703716. doi: 10.3934/dcds.2008.21.703 
[18] 
Hyungjin Huh. Selfsimilar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 5360. doi: 10.3934/eect.2018003 
[19] 
Thomas Y. Hou, Ruo Li. Nonexistence of locally selfsimilar blowup for the 3D incompressible NavierStokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637642. doi: 10.3934/dcds.2007.18.637 
[20] 
K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. selfsimilar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 5176. doi: 10.3934/cpaa.2002.1.51 
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]