Multimodal Maps Research Articles

Non-uniform hyperbolicity in complex dynamics

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n c so that $\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty$ and F has no parabolic periodic cycles. Let μ max be the maximal multiplicity of the critical points. The objective is to study the Poincare series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent $\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}$ where δ Poin (J) is the Poincare exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ Poin (J)=HDim(J). If F is polynomially summable with the exponent α, $\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty$ and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval. We prove that if F is summable with an exponent $\alpha< \frac{2}{2+\mu_{\textit{max}}}$ then the Minkowski dimension of J is strictly less than 2 if $J\neq\hat{\mathbb{C}}$ and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with $\alpha<\frac{1}{1+\mu_{\textit{max}}}$ then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability. Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.

Multisensory Integration: A Functional Role for Inter-Area Synchronization?

It has been suggested that synchronization of neural activity in distinct areas of the brain is important in the 'perceptual binding' of the various features relating to a specific object. A recent study has provided new evidence that inter-area synchronization is important in multisensory integration.

Inhibition, not excitation, is the key to multimodal sensory integration

Multimodal neuronal maps, combining input from two or more sensory systems, play a key role in the processing of sensory and motor information. For such maps to be of any use, the input from all participating modalities must be calibrated so that a stimulus at a specific spatial location is represented at an unambiguous position in the multimodal map. Here we discuss two methods based on supervised spike-timing-dependent plasticity (STDP) to gauge input from different sensory modalities so as to ensure a proper map alignment. The first uses an excitatory teacher input. It is therefore called excitation-mediated learning. The second method is based on an inhibitory teacher signal, as found in the barn owl, and is called inhibition-mediated learning. Using detailed analytical calculations and numerical simulations, we demonstrate that inhibitory teacher input is essential if high-quality multimodal integration is to be learned rapidly. Furthermore, we show that the quality of the resulting map is not so much limited by the quality of the teacher signal but rather by the accuracy of the input from other sensory modalities.

Lifting measures to inducing schemes

AbstractIn this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

Large derivatives, backward contraction and invariant densities for interval maps

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfn(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to Lp for all p<ℓmax/(ℓmax-1), where ℓmax denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.

Equilibrium measures for maps with inducing schemes

We introduce a class of continuous maps f of a compact metric space I admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamical formalism, i.e., describe a class of real-valued potential functions \phi on I which admit unique equilibrium measures \mu_\phi minimizing the free energy for a certain class of measures. We also describe ergodic properties of equilibrium measures including decay of correlation and the Central Limit Theorem. Our results apply in particular to some one-dimensional unimodal and multimodal maps as well as to multidimensional nonuniformly hyperbolic maps admitting Young's tower. Examples of potential functions to which our theory applies include \phi_t=-t\log|df| with t\in(t_0, t_1) for some t_0<1<t_1. In the particular case of S-unimodal maps we show that one can choose t_0<0 and that the class of measures under consideration comprises all invariant Borel probability measures. Thus our results establish existence and uniqueness of both the measure of maximal entropy (by a different method than Hofbauer) and the absolutely continuous invariant measure extending results by Bruin and Keller for the parameters under consideration.

Visible measures of maximal entropy in dimension one

A real one-dimensional analogue of Zdunik’s dichotomy is proven giving dynamical conditions for a multimodal map to have a measure of maximal entropy of dimension one.

C*-algebras associated with interval maps

For each piecewise monotonic map τ \tau of [ 0 , 1 ] [0,1] , we associate a pair of C*-algebras F τ F_\tau and O τ O_\tau and calculate their K-groups. The algebra F τ F_\tau is an AI-algebra. We characterize when F τ F_\tau and O τ O_\tau are simple. In those cases, F τ F_\tau has a unique trace, and O τ O_\tau is purely infinite with a unique KMS state. In the case that τ \tau is Markov, these algebras include the Cuntz-Krieger algebras O A O_A , and the associated AF-algebras F A F_A . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and β \beta -transformations. For the case of interval exchange maps and of β \beta -transformations, the C*-algebra O τ O_\tau coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

Erratum to “Real bounds, ergodicity and negative Schwarzian for multimodal maps”

A technical assumption in Part 1 of Theorem C of the authors’ article Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc. 17 (2004), 749–782, was, by mistake, omitted. Here we explain that the conclusion of the theorem holds if the interval we pullback is “nice”.

Computer-assisted 3D ultrasound-guided neurosurgery: technological contributions, including multimodal registration and advanced display, demonstrating future perspectives

Navigation systems are now frequently being used for guiding surgical procedures. Existing neuronavigation systems suffer from the lack of updated images when tissue changes during surgery as well as from user-friendly displays of all essential images for accurate and safe surgery guidance. We have developed various new technologies for improved neuronavigation. Using intraoperative 3D ultrasound (US) imaging, we have developed various registration algorithms for using and updating a complete multimodal and multivolume 3D map for navigation. We experienced that advanced multimodal visualization makes it easy to interpret information from several image volumes and modalities simultaneously. Using high quality intraoperative 3D ultrasound, essential preoperative information could be corrected due to brain shift. fMRI and other important preoperative data could then be used together with intraoperative ultrasound imaging for more accurate, safer and improved guidance of therapy. We claim that new features, as demonstrated in the present paper, using intraoperative 3D ultrasound in combination with advanced registration and display algorithms will represent important contributions towards more accurate, safer and more optimized future patient treatment.

Topological invariance of generic non-uniformly expanding multimodal maps

We show that a strengthened version of the Collet-Eckmann condition for multimodal maps is topologically invariant. In particular, if f is non-uniformly expanding and the critical points are generic with respect to the absolutely continuous invariant measure, then any map g topologically conjugate to f is also non-uniformly expanding.

Phase space universality for multimodal maps

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.

Order isomorphism under higher power shift and generic form of renormalization group equations in multimodal maps

A framework of the sequence renormalization in symbolic space, which degenerates for order patterns of block shifts, is presented in this paper. The order isomorphism of both the higher power shift on a block sequence and the shift on a sequence is the topological and metric foundation of renormalization group (RG) operator. Based on the order relation with many compositions, the generic form of RG equations in multimodal maps is derived.

Multimodal trivariate thematic maps with auditory and haptic display

AbstractThe purpose of this study is to explore the possibility of multimodal trivariate thematic maps by utilizing auditory and haptic displays. With two different modes of display (color only displays and multimodal displays with color, audio, and haptic), the completion time of tasks and the recall (retention) rate were measured in two separate experiments. In terms of the completion time, with minimal learning experience, participants could accomplish the tasks at the same level of speed in both modes. For the recall rate, multimodal displays have much higher recall rates. It is speculated that the natural quantitative hierarchies in auditory and haptic displays provide an advantage in the use of multimodal displays. These findings confirmed the possibility of using auditory and haptic displays in visually dominant geographic information systems (GIS) for multi‐variate thematic maps.

Supporting Multi-Modal Interfaces for Adaptive Tour Planning

Typical examples for mobile GI services include tour planning and maps for navigation support (Reuter, Zipf, 2004). We have developed several prototypes of mobile GI services for navigation tasks including innovative services like user-adaptive tour planning. We are also among the first to support multi-modal (i.e. graphical, natural language and gestures based) user interfaces for such GI related mobile services in several projects like Deep Map, Crumpet and SmartKom since several years. As the development of GI services that adapt to user and context parameters as well as multi-modal interfaces for mobile navigation support are relative new research areas for GIScience it was necessary to specify interfaces for the software components that have been developed. These–relevant parts of the so-called Deep Map Objects (DMO), the XML-based message objects of the Deep Map multi agent system–will be introduced in this paper. But only very recently also the OGC Open Location Services Initiative (OpenLS) has published a specification for similar services in the area of LBS–but yet without a focus on adaptation or multi-modality. Both representations include relevant data types for route planning and map generation and are based on XML. Therefore we will compare these two representations and discuss their applicability for developing multi-modal map interaction for adaptive tour planning.

Chaotic Coexistence in a Top-predator Mediated Competitive Exclusive web

A basic food web of 4 species is considered, of which there is a bottom prey X, two predators Y, Z on X, and a top predatorW only on Y. The study concerns with how one type of chaotic coexistence arises. It is shown that under the situation that without the top-predator W, competitor Z goes to extinction, without Z the XYW locks in a periodic cycle, yet with all species, the noncompetitiveZ can derive the dynamics from periodic orbits to chaos. The dynamics can be captured analytically by 1-dimensional unimodal and multimodal maps and symbolic shift maps.

Order isomorphism under higher power shift and generic form of renormalization group equations in multimodal maps

Abstract A framework of the sequence renormalization in symbolic space, which degenerates for order patterns of block shifts, is presented in this paper. The order isomorphism of both the higher power shift on a block sequence and the shift on a sequence is the topological and metric foundation of renormalization group (RG) operator. Based on the order relation with many compositions, the generic form of RG equations in multimodal maps is derived.

Real bounds, ergodicity and negative Schwarzian for multimodal maps

We consider smooth multimodal maps which have finitely many non-flat critical points. We prove the existence of real bounds. From this we obtain a new proof for the non-existence of wandering intervals, derive extremely useful improved Koebe principles, show that high iterates have ‘negative Schwarzian derivative’ and give results on ergodic properties of the map. One of the main complications in the proofs is that we allow f f to have inflection points.

Multimodal map and complex basin of attraction of a simple hopper.

In this paper, we study the global dynamics of a simple passive mechanical model for hopping. The hopper is a two-mass, single-spring system constrained to move in the vertical direction (under gravity) above a rigid ground. The hopper model and its basic dynamics including the existence of incessant hopping motions have been reported elsewhere. Here, we extend the study to investigate the global dynamics of the hopper. The global map of the hopper is multimodal. We construct an approximate analytic map near the fixed points of the map and show that the fixed points exhibit one-way stability. We also show that the map is invariant under the inversion of the mass ratio of the hopper. Next, we construct the global basin of attraction of these fixed points and show that their structure is highly complex and retains form at finer scales. This structure of the basin of attraction contains regions where the fate of an arbitrary initial condition becomes unpredictable.

An effective numerical method of the word-lifting technique in one-dimensional multimodal maps

For general algebraic maps of orders higher than or equal to 5, the word-lifting technique for calculating parameters of symbolic sequences meets with a difficulty of lacking explicit expressions of the inverse functions of the maps. This Letter presents an effective numerical method to solve such a problem. Parameters of some quadruply superstable kneading sequences in quadrumodal quintic maps are calculated as examples.