Isotypes of ICIS and images of deformations of map germs

We give a simple way to study the isotypes of the homology of simplicial complexes with actions of finite groups, and use it for Milnor fibers of \textsc{icis}. We study the homology of images of mappings $f_t$ that arise as deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with $n<p$, and the behaviour of singularities (instabilities) in this context. We study two generalizations of the notion of image Milnor number $\mu_I$ given by Mond and give a workable way of compute them, with Milnor numbers of \textsc{icis}. We also study two unexpected traits when $p>n+1$: stable perturbations with contractible image and homology of $\text{im} f_t$ in unexpected dimensions. We show that Houston's conjecture, $\mu_I$ constant in a family implies excellency in Gaffney's sense, is false, but we give a proof for the cases where it holds. Finally, we prove a result on coalescence of instabilities for the cases $(n,p)=(1,2), (2,3)$, showing that it is false in general.Comment: 38 pages, 11 figure

A note on complex plane curve singularities up to diffeomorphism and their rigidity

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities

Chaotic differential operators

We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space lp, where B is the backward shift operator. © 2011 Springer-Verlag.This work was partially supported by the MEC and FEDER Projects MTM2007-64222, MTM2010-14909, and by GVA Project GV/2010/091, and by UPV Project PAID-06-09-2932. The authors would like to thank A. Peris for helpful comments and ideas that produced a great improvement of the paper's presentation. We also thank the referees for their helpful comments and for reporting to us a gap in Theorem 1.Conejero Casares, JA.; Martínez Jiménez, F. (2011). Chaotic differential operators. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 105(2):423-431. https://doi.org/10.1007/s13398-011-0026-6S4234311052Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bermúdez T., Miller V.G.: On operators T such that f(T) is hypercyclic. Integr. Equ. Oper. Theory 37(3), 332–340 (2000)Bonet J., Martínez-Giménez F., Peris A.: Linear chaos on Fréchet spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7), 1649–1655 (2003)Chan K.C., Shapiro J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40(4), 1421–1449 (1991)Conejero J.A., Müller V.: On the universality of multipliers on $${\mathcal{H}({\mathbb {C}})}$$ . J. Approx. Theory. 162(5), 1025–1032 (2010)deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dyn. Syst. 21(5), 1411–1427 (2001)Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. In: Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.,: Linear chaos. Universitext, Springer, New York (to appear, 2011)Herzog G., Schmoeger C.: On operators T such that f(T) is hypercyclic. Stud. Math. 108(3), 209–216 (1994)Kahane, J.-P.: Some random series of functions, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)Martínez-Giménez F., Peris A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(8), 1703–1715 (2002)Martínez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)Salas H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 93–1004 (1995)Shapiro, J.H.: Simple connectivity and linear chaos. Rend. Circ. Mat. Palermo. (2) Suppl. 56, 27–48 (1998

Linear chaos for the Quick-Thinking-Driver model

The final publication is available at Springer via http://dx.doi.org/10.1007/s00233-015-9704-6In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars (Google driverless cars, http://en.wikipedia.org/wiki/Google_driverless_car).Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behaviour using the same approach as for studying chaos of death models of cell growth.The authors were supported by a grant from the FPU program of MEC and MEC Project MTM2013-47093-P.Conejero, JA.; Murillo Arcila, M.; Seoane-Sepúlveda, JB. (2016). 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A weak version of Mond's conjecture

We prove that a map germ $f:(\mathbb{C}^n,S)\to(\mathbb{C}^{n+1},0)$ with isolated instability is stable if and only if $\mu_I(f)=0$, where $\mu_I(f)$ is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that $f$ has corank one. The proof here is also valid for corank $\ge 2$, provided that $(n,n+1)$ are nice dimensions in Mather's sense (so $\mu_I(f)$ is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the $\mathcal{A}_e$-codimension of $f$ is $\le \mu_I(f)$, with equality if $f$ is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of $f$ is a hypersurface.Comment: 18 pages, 5 figure