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.
We prove that on any surface there is a C^\infty diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe \Lambda which is equivalent to the unique non-trivial Hausdorff measure in \Lambda. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe \Lambda are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.
Abstract. We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.
Toward improved prediction of the bedrock depth underneath hillslopes: Bayesian inference of the bottom-up control hypothesis using high-resolution topographic data Abstract The depth to bedrock controls a myriad of processes by influencing subsurface flow paths, erosion rates, soil moisture, and water uptake by plant roots. As hillslope interiors are very difficult and costly to illuminate and access, the topography of the bedrock surface is largely unknown. This essay is concerned with the prediction of spatial patterns in the depth to bedrock (DTB) using high-resolution topographic data, numerical modeling, and Bayesian analysis. Our DTB model builds on the bottom-up control on freshbedrock topography hypothesis of Rempe and Dietrich (2014) and includes a mass movement and bedrock-valley morphology term to extent the usefulness and general applicability of the model. We reconcile the DTB model with field observations using Bayesian analysis with the DREAM algorithm. We investigate explicitly the benefits of using spatially distributed parameter values to account implicitly, and in a relatively simple way, for rock mass heterogeneities that are very difficult, if not impossible, to characterize adequately in the field. We illustrate our method using an artificial data set of bedrock depth observations and then evaluate our DTB model with real-world data collected at the Papagaio river basin in Rio de Janeiro, Brazil. Our results demonstrate that the DTB model predicts accurately the observed bedrock depth data. The posterior mean DTB simulation is shown to be in good agreement with the measured data. The posterior prediction uncertainty of the DTB model can be propagated forward through hydromechanical models to derive probabilistic estimates of factors of safety.
In tropical areas, mass movements are common phenomena, especially during periods of heavy rainfall, which frequently take place in the summer season. These phenomena have caused loss of life and serious damage to infrastructure and properties. The most prominent of these phenomena are landslides that can produce debris flows. Thus, this article aims at determining affected areas using a model to predict landslide prone areas (SHALSTAB) combined with an empirical model designed to define the debris flow travel distance and area of deposition. The methodology of this work consists of the following steps: (a) elaboration of a digital elevation model (DEM), (b) application of the deterministic SHALSTAB model to locate the landslide prone areas, (c) identification of the debris flow travel distance and area of deposition, and (d) mapping of the affected areas (landslides and debris flows). This work was developed in an area in which many mass movements occurred after intense rainfall during the summer season (February 1996) in the state of Rio de Janeiro, southeast Brazil. All of the scars produced by that event were mapped, allowing for validation of the applied models. The model results show that the mapped landslide locations can adequately be simulated by the model.
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