Marcelo Freitas Caetano
email: caetano@ircam.fr

Xavier Rodet
email: rodet@ircam.fr

AUTOMATIC TIMBRAL MORPHING OF MUSICAL INSTRUMENT SOUNDS BY HIGH-LEVEL DESCRIPTORS

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The 20 th century witnessed a compositional paradigm shift from pitch and duration to timbre [START_REF] Wishart | On Sonic Art[END_REF]. The advent of the digital computer revolutionized the representation and manipulation of sounds, opening up new avenues of exploration. Timbre manipulation led to the development of transformational techniques usually referred to as morphing. Among the several possible applications of morphing [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], the exploration of the sonic continuum in composition [START_REF] Wishart | On Sonic Art[END_REF] stands out as the most exciting to date. Jonathan Harvey's 'Mortuos plango, vivos voco' morphs seamlessly from a vowel sung by a boy to the complex bell spectrum consisting of many partials. Another example is Trevor Wishart's Red bird where the word 'listen' gradually morphs into birdsong [START_REF] Wishart | On Sonic Art[END_REF]. Wishart himself mentions Michael McNabb's 'Dreamsong' and its particularly striking opening and closing morphs [START_REF] Wishart | [END_REF]. These authors did morphing by hand mainly using studio techniques. This work investigates techniques to automatically achieve similar results by simply choosing what sounds we want to morph between and how we want to do the transformation, especially because many different transformations fall under the umbrella of morphing, as we will explain in more detail in Section 2. There seems to be no consensus on what sound morphing is. Most authors seem to agree that morphing involves the hybridization of two (or more) sounds by blending auditory features. One frequent requirement is that the result should fuse into a single percept, somewhat ruling out simply mixing the sources [START_REF] Fitz | Sinusoidal Modeling and Manipulation Using Lemur[END_REF], [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], because the ear is still usually capable of distinguishing them due to a number of cues and auditory processes. Sill, many different transformations are described as morphing, such as interpolated timbres [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], smooth, seamless transitions between sounds [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF] or cyclostationary morphs [START_REF] Slaney | Automatic Audio Morphing[END_REF], each of which will be thoroughly reviewed in Section 2. Most authors propose to interpolate the parameters of a model [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF], [2], [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF], [START_REF] Hope | Endemic Problems in Timbre Morphing Processes: Causes and Cures[END_REF], [START_REF] Osaka | Timbre Interpolation of Sounds Using a Sinusoidal Model[END_REF], [START_REF] Roebel | Morphing Dynamical Sound Models[END_REF], [START_REF] Slaney | Automatic Audio Morphing[END_REF], [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF] without worrying about the perceptual impact of the process. These authors often conclude that the linear interpolation of the parameters do not correspond to linearly varying the corresponding features [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF], [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF], [START_REF] Slaney | Automatic Audio Morphing[END_REF]. Some authors proposed timbre spaces [START_REF] Grey | Perceptual Evaluations of Synthesized Musical Instrument Tones[END_REF], [START_REF] Caclin | Acoustic Correlates of Timbre Space Dimensions: Study Using Synthetic Tones[END_REF], where each dimension is correlated to a perceptual feature. Caetano [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF] figures among the first to make a distinction between interpolation of parameters and morphing of features. Our motivation is the hybridization of perceptual features of musical instrument sounds that are related to salient timbral dimensions unveiled in psychoacoustic experiments [START_REF] Caclin | Acoustic Correlates of Timbre Space Dimensions: Study Using Synthetic Tones[END_REF], [START_REF] Krimphoff | Caractérisation du Timbre des sons Complexes. II: Analyses Acoustiques et Quantification Psychophysique[END_REF], [START_REF] Mcadams | Perceptual Scaling of Synthesized Musical Timbres: Common Dimensions, Specifities and Latent Subject Classes[END_REF]. In other words, instead of simply obtaining hybrid sounds, we want to control the hybridization process perceptually. In this work, we describe techniques to automatically obtain perceptually intermediate quasi-harmonic musical instrument sounds using high-level descriptors as guides. High level descriptors are measures of acoustic correlates of timbre dimensions obtained by perceptual studies, such that sounds whose features are intermediate between two would be placed between them in the underlying timbre space used as guide.

The next section contains a comprehensive review of the terminology and processes usually called morphing, followed by the techniques proposed to achieve the desired results. Next, we briefly review timbre perception and timbre spaces, and introduce high-level descriptors. Then, we propose a timbral morphing technique that consists of extracting the features, interpolating between them in the descriptor domain, thought to capture perceptual timbral features, and resynthesizing the morphed sound with parameter values that correspond to the morphed features. We emphasize methods to obtain a morphed spectral envelope with hybrid descriptor values. Finally, we present the conclusions and future perspectives of the morphing technique.

WHAT IS SOUND MORPHING?

After a thorough review of the literature on the hybridization of sounds, we realized there is much confusion in terminology. One of the aims of this article is to clarify a little bit the techniques referred to as morphing and the terminology itself used in the literature. Apart from sound morphing, some authors refer to this hybridization process as audio morphing [START_REF] Slaney | Automatic Audio Morphing[END_REF], while others prefer timbre morphing [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF] or even timbre interpolation [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF] to refer to similar goals, and some choose to use these terms interchangeably. The result has been called hybrid [START_REF] Haken | Beyond Traditional Sampling Synthesis: Real-Time Timbre Morphing Using Additive Synthesis[END_REF] [7], intermediate [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF], interpolated [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF] or even mongrel sound [START_REF] Hope | Endemic Problems in Timbre Morphing Processes: Causes and Cures[END_REF]. In this work, we reserve the term sound for the auditory impression or the sensation perceived by the sense of hearing, whereas audio refers more specifically to the signal. Moreover, we make a distinction between interpolation and morphing. Interpolation acts on the parameters of a model, being restricted to the signal level, whereas we reserve morphing for the hybridization of perceptual qualities. So we propose sound morphing as the most appropriate term to our goals, and we talk about hybrid or intermediate sounds. We focus on timbral features independent from loudness and pitch (LP-timbre, as defined by Letowsi [START_REF] Letowski | Tone Color, and Sound Quality: Concepts and Definitions[END_REF]), especially those related to the spectral envelope shape [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF], so we will make an additional important distinction between timbre morphing and the term we chose to use here, timbral morphing, while attempting to find a good definition for sound morphing. There seems to be no widely accepted definition of morphing in the literature. Instead, most authors either attempt to provide a definition of their own or simply explain what the aim of their work was. Some definitions are too system dependent to be useful, Fitz [START_REF] Fitz | Sinusoidal Modeling and Manipulation Using Lemur[END_REF] defines morphing as "the process of combining two or more Lemur files to create a new Lemur file with an intermediate timbre", others are too general, such as Boccardi's [2] "modifying the time-varying spectrum of a source sound to match the time-varying spectrum of a given number of target sounds". Definitions based on the concept of timbre are common [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF], [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], [START_REF] Osaka | Concatenation and Stretch/Squeeze of Musical Instrumental Sound Using Morphing[END_REF], [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF]. Usually, these authors define timbre morphing as "the process of combining two or more sounds to create a new sound with intermediate timbre" [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF] or "to achieve a smooth transition from one timbre to another" [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF]. We should notice that these refer to different goals. All in all, we prefer to avoid any definition that relies heavily on a concept as loosely defined and misunderstood as timbre, that can encompass many different perceptual dimensions of sounds [START_REF] Letowski | Tone Color, and Sound Quality: Concepts and Definitions[END_REF]. Although nobody defines what they mean by timbre, most authors seem to refer to timbre as the set of attributes that allow sound source identification. In musical instrument contexts, this usually means that timbre becomes a synonym of musical instrument and thus timbre morphing reduces to hybrid musical instrument sounds. It is possible, though, to morph between sounds from the same instrument (different loudness or even different temporal features) [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], [START_REF] Slaney | Automatic Audio Morphing[END_REF]. Instead, we prefer to define the aim of morphing as obtaining a sound that is perceptually intermediate between two (or more), such that our goal becomes to hybridize perceptually salient features of sounds related to timbre dimensions, which we term timbral morphing.

Slaney [START_REF] Slaney | Automatic Audio Morphing[END_REF], on the other hand, prefers to avoid a direct definition altogether and explains the concept by analogy with image morphing instead, where the aim is to gradually change from one image (the source) to the other (the target) producing convincing intermediates (or hybrids) along the way. Other authors have proposed the same analogy [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF]. Nonetheless, they rely on the concept of sound object especially because they do not restrict their goal to musical instrument sounds. Figure 1 shows such an example of image morphing with faces.

Clearly, it is not enough to blindly interpolate parameters (pixels, for instance, for the images) since there are a number of important features in the faces that we must take into account. Finding those features is an important task, and developing techniques to obtain intermediate (hybrid) images that use those features as cues is the key to a successful morph. Here we argue that high-level descriptors capture salient timbre dimensions of sounds, so we use them to align temporal features and to morph spectral shapes. An important concept that can be inferred from Figure 1 is the fact that there are many possible intermediate steps between the two images shifting from the source to the target. The original images/sounds from now on shall arbitrarily be called source and target for formalization purposes only because the morph should not be different if they change positions. So, if we consider each intermediate image/sound as the result of a different combination of source and target, this convex combination can be mathematically expressed as equation ( 1) and each step is characterized by one value of a single parameter (α), called interpolation or morphing factor, as shown at the bottom of Figure 1. The morphing factor should vary between 0 and 1, such that α = 1 and α = 0 produce source and target respectively. Convex combinations of more than two objects (images, sounds) are also possible, as well as using a time varying morphing factor, giving rise to dynamic transformations.

( ) [ ] 2 1 ) ( 1 ) ( , S t S t t M α α α - + = (1)
Due to the intrinsic temporal nature of sounds, a better analogy would be that of movie morphing [START_REF] Slaney | Automatic Audio Morphing[END_REF], where the aim must be reviewed to better fit the dynamic nature of the media, depicted in Figure 2. Now our sound morphing analogy has closer correspondences. For example, each movie frame could correspond to an STFT frame resulting from the analysis of the sounds we intend to morph between. Also, we can imagine that each frame's visual features have a corresponding set of sonic features that also evolve in time and that this evolution in time itself carries important information about how we perceive the movie (sound). Notice that Figure 2 depicts movies (sounds) with different numbers of frames, therefore, different lengths (supposing the same frame rate). This is a somewhat trickier problem than image morphing because of the added temporal dimension. Now we need to choose what kind of transformation we intend to do. We could simply make a movie that contains an intermediate number of frames, but we need to account for important temporal information to make it more convincing. If the first movie shows an explosion at the beginning (similarly to the abrupt attack of a plucked string or a percussive sound) and the other a butterfly gently flapping its wings and then flying away, we might need to align relevant temporal cues to produce an interesting morph. Moreover, there are a number of possible transitions between the two. Do we want an intermediate movie that contains morphed images of each frame (here called static or stationary morphing because α is constant), or are we going for a movie that starts as the first and dynamically changes into the other (here called dynamic morphing because α varies in time)?

We could choose to run the first frames of the first movie until we stop at a selected frame, gradually morph it into another selected frame of the second, and then proceed by showing the rest of it (warped dynamic morphing), choosing to somehow warp the length of the result in order to achieve a given effect. Finally, another possibility would be to produce several hybrid sounds in different intermediate points (i.e., different values of α) of the path between source and target (cyclostationary morphing). With these considerations in mind, a world of possibilities opens up, from the trajectory followed by the morph determined by α to the choice of source and target sounds to be morphed between. We just need to bear in mind that all these choices affect the quality of the results and might even be somewhat intertwined. For instance, it might be easier to morph between a trumpet note and the singing voice than drums.

MORPHING SOUNDS

The aim of this section is to review the morphing techniques and highlight the aim of using descriptors to guide the transformation. Most morphing techniques proposed in the literature consist in describing a model used to analyse the sounds and interpolating the parameters of the model regardless of features [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF], [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF], [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], [START_REF] Fitz | Sinusoidal Modeling and Manipulation Using Lemur[END_REF], [2], [START_REF] Osaka | Concatenation and Stretch/Squeeze of Musical Instrumental Sound Using Morphing[END_REF], [START_REF] Osaka | Timbre Interpolation of Sounds Using a Sinusoidal Model[END_REF], [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF]. The basic idea behind the interpolation principle is that if we can represent different sounds by simply adjusting the parameters of a model, we should obtain a somewhat smooth transition between two (or more) sounds by interpolating between these parameters. Interpolation of sinusoidal modelling is amongst the most common approaches [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF], [2], [START_REF] Hatch | High-Level Audio Morphing Strategies[END_REF], [START_REF] Osaka | Concatenation and Stretch/Squeeze of Musical Instrumental Sound Using Morphing[END_REF], [START_REF] Osaka | Timbre Interpolation of Sounds Using a Sinusoidal Model[END_REF], [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], [START_REF] Williams | Perceptually-Motivated Audio Morphing: Softness[END_REF], [START_REF] Wishart | [END_REF]. Tellman [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF] offers us one of the earliest descriptions of a morphing technique, which is based on a sinusoidal representation [START_REF] Fitz | Sinusoidal Modeling and Manipulation Using Lemur[END_REF]. The morphing scheme consists of interpolating the result of the Lemur [START_REF] Fitz | Sinusoidal Modeling and Manipulation Using Lemur[END_REF] analysis and involves time-scale modification to morph between different attack and vibrato rates. More recently, Fitz [START_REF] Fitz | Cell-Utes and Flutter-Tongued Cats: Sound Morphing Using Loris and the Reassigned Bandwidth-Enhanced Model[END_REF] presented a morphing technique also using a sinusoidal representation, and morphing is achieved again by simply interpolating the parameters of the model. Hope [START_REF] Hope | Endemic Problems in Timbre Morphing Processes: Causes and Cures[END_REF], [START_REF] Hope | Time-frequency Distributions for Timbre Morphing: the Wigner distribution versus the STFT[END_REF] prefers to interpolate the parameters of a Wiegner Distribution analysis. Boccardi [2], in turn, uses GMM to interpolate between additive parameters (SMS) [START_REF] Serra | Musical Sound Modeling with Sinusoids Plus Noise[END_REF]. Röbel [START_REF] Roebel | Morphing Dynamical Sound Models[END_REF] proposes to model sounds as dynamical systems with neural networks and to morph them by interpolating the attractors corresponding to those dynamical systems. Ahmad [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF] applies a discrete wavelet transform and singular value decomposition to morph between transient sounds. They interpolate linearly between the parameters and state that other interpolation strategies with a better perceptual correlation should be studied.

A few authors have proposed to detach the spectral envelope from the frequency information and interpolate them separately [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF], [START_REF] Ezzat | Morphing Spectral Envelopes using Audio Flow[END_REF], [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF], [START_REF] Slaney | Automatic Audio Morphing[END_REF]. Slaney [START_REF] Slaney | Automatic Audio Morphing[END_REF] proposes to represent the sounds to be morphed between in a multidimensional space that encodes spectral shape and pitch in orthogonal axes and warp the dimensions of this representation to obtain the desired result. However, they represent spectral shape by MFCCs and pitch information by a residual spectrogram calculation, which are then interpolated using dynamic time warping and harmonic alignment as guides. They conclude by stating that the method should be improved with perceptually optimal interpolation functions. Ezzat [START_REF] Ezzat | Morphing Spectral Envelopes using Audio Flow[END_REF] uses a spectral smoothing technique to morph spectral envelopes. They analyse soberly the problem of interpolating spectral envelopes and argue that this approach accounts for proper formant shifting between source and target. We shall verify this claim in Section 6, and also verify if it accounts for the morphing of timbral features as a perceptually motivated morphing algorithm should. Finally, only recently did we start to take perceptual aspects into consideration [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF], [START_REF] Williams | Perceptually-Motivated Audio Morphing: Softness[END_REF], [START_REF] Williams | Perceptually-Motivated Audio Morphing: Brightness[END_REF], [START_REF] Hatch | High-Level Audio Morphing Strategies[END_REF], and the result is the addition of one more step in the process, feature calculation. In most models proposed, linear variation of interpolation parameter does not produce perceptually linear morphs [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF], so recently authors have started to study the perceptual impact of their models and how to interpolate the parameters so that the results vary roughly linearly on the perceptual sphere. Williams [START_REF] Williams | Perceptually-Motivated Audio Morphing: Softness[END_REF], [START_REF] Williams | Perceptually-Motivated Audio Morphing: Brightness[END_REF] studies an additive-based perceptuallymotivated technique to morph sounds guided by the spectral centroid. They selectively amplify or attenuate harmonics of sawtooth or square waves to tilt the centroid towards that of the target sound. Hikichi [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF] uses MDS spaces [START_REF] Mcadams | Perceptual Scaling of Synthesized Musical Timbres: Common Dimensions, Specifities and Latent Subject Classes[END_REF] constructed from the sources and morphed sounds to figure out how to warp the interpolation factor in the parameter space so that it will linearly morph in the perceptual domain. Hatch [START_REF] Hatch | High-Level Audio Morphing Strategies[END_REF] poses the problem of feature interpolation very clearly but it remains unclear how he matches target values of spectral centroid, for example. Caetano [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF] proposes to morph spectral envelopes guided by descriptors controlling the spectral shape by changing the parameters of the spectral envelope model with the aid of a genetic algorithm. In this work, we are going to present strategies to achieve perceptually relevant morphing of quasi-harmonic musical instrument sounds taking most temporal and spectral timbral aspects of sounds into account. Particularly in the spectral domain, we are looking for a spectral envelope representation that best approximates linear interpolation on the perceptual timbre space when we linearly interpolate the parameters. We use quasi-harmonic musical instrument sounds so that the partials have a simple correspondence, although the techniques herein described could easily be extrapolated to vocal sounds (singing voice) or inharmonic sounds.

ACOUSTIC CORRELATES OF TIMBRE SPACES

In this section we briefly present timbre perception, timbre spaces and the most relevant acoustic correlates of timbral dimensions obtained in the literature of timbre perception. The concept of timbre is related to the subjective response to the perceptual qualities of sound objects and events [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. We know that source identification is not reduced to waveform memorization because the intrinsic dynamic nature of the sources produces variations [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. Timbre perception is inherently multidimensional, involving features such as the attack, spectral shape, and harmonic content. Since the pioneering work of Helmholtz [START_REF] Helmholtz | On the Sensations of Tone[END_REF], multidimensional scaling techniques figure among the most prominent when trying to quantitatively describe timbre. McAdams [START_REF] Mcadams | Perceptual Scaling of Synthesized Musical Timbres: Common Dimensions, Specifities and Latent Subject Classes[END_REF] gives a comprehensive review of the early timbre space studies. Grey [START_REF] Grey | Perceptual Evaluations of Synthesized Musical Instrument Tones[END_REF] investigated the multidimensional nature of the perception of musical instrument timbre, constructed a three-dimensional timbre space, and proposed acoustic correlates for each dimension. He concluded that the first dimension corresponded to spectral energy distribution (spectral centroid), the second and third dimensions were related to the temporal variation of the notes (onset synchronicity). Krumhansl [START_REF] Krumhansl | Why is Musical Timbre So Hard to Understand?[END_REF] conducted a similar study using synthesized sounds and also found three dimensions related to attack, synchronicity and brightness. Krimphoff [START_REF] Krimphoff | Caractérisation du Timbre des sons Complexes. II: Analyses Acoustiques et Quantification Psychophysique[END_REF] studied acoustic correlates of timbre dimensions and concluded that brightness is correlated with the spectral centroid and rapidity of attack with rise time in a logarithmic scale. McAdams [START_REF] Mcadams | Perceptual Scaling of Synthesized Musical Timbres: Common Dimensions, Specifities and Latent Subject Classes[END_REF] conducted similar experiments with synthesized musical instrument timbres and concluded that the most salient dimensions were log rise time, spectral centroid and degree of spectral variation.

More recently, Caclin [START_REF] Caclin | Acoustic Correlates of Timbre Space Dimensions: Study Using Synthetic Tones[END_REF] studied the perceptual relevance of a number of acoustic correlates of timbre-space dimensions with MDS techniques and concluded that listeners use attack time, spectral centroid and spectrum fine structure in dissimilarity rating experiments.

Listeners use many acoustical properties to identify events, such as the spectral shape, formant frequencies, attack (onset) and decay (offset), noise, among others [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. The cues to identification and timbre vary across notes, durations, intensities and tempos [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. One model of sound production is based on two possibly interactive components, the source and the filter [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. The basic notion is that the source is excited by energy to generate a vibration pattern composed of several vibration modes (modelled as sinusoidal components). This pattern is imposed on the filter, which acts to modify the relative amplitudes of the components of the source input [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. We obtain estimates of the filter by calculating the spectral envelope, which is a smooth curve that approximately matches the peaks of the spectrum. The peaks of the spectral envelope (also called formants in voice research) correspond roughly to the vibration modes of the sourcefilter model. The number and absolute position of spectral peaks in frequency is important for musical instrument (sound source) identification and here we refer to it as spectral form to distinguish from the spectral shape, which is correlated with dimensions of timbre spaces obtained from perceptual studies. We note that envelope form and shape complement each other, since there are several possible spectral envelopes with different forms and the same shape, i.e., values of descriptors. So we say that to obtain perceptually intermediate spectral envelopes we need not only to take spectral form but also spectral shape into account. In other words, we need to obtain a spectral envelope with an intermediate number and absolute position of formant peaks and also intermediate brightness (centroid), roughness (spread), etc. Obtaining an intermediate spectral shape corresponds to placing the sounds between two (or more) in the corresponding underlying timbre space that generated the dimensions. Supposing that timbre space is orthogonal (like in MDS studies), then intermediate points in high-dimensional space have intermediate values for each dimension (that is, intermediate descriptors), as illustrated on the left-hand side of Figure 3. We see a two-dimensional orthogonal abstraction of timbre space where each dimension corresponds to a feature captured by a descriptor. We also see two sound objects represented by the circle and the square and their corresponding features reflected as the values of the descriptors on each axis. The intermediate sound object represented by the square with rounded corners must have intermediate features, and therefore intermediate values of descriptors.

HIGH-LEVEL DESCRIPTORS

We measure timbral features with high-level descriptors, such that a sound with intermediate descriptors should be perceived as intermediate. We adopted temporal and spectral features in our study to account for prominent timbre dimensions. The temporal features are log attack and decay times, energy (temporal) envelope, and temporal evolution of harmonic contents, usually referred to as shimmer and jitter. The spectral features are form (formant peaks) and shape (centroid, spread, skewness, kurtosis and slope). Notice that the spectral features are extracted from both the sinusoidal and noise components of the analysis. In this section we present the general scheme used to calculate all he descriptors used in this work, depicted in Figure 4. The sound signal is highlighted with a dark background, all the purely signal processing stages have white background and the steps where we calculate the descriptors present a light background. Peeters [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF] describes exhaustively how to calculate all the descriptors we use in this work and proposes to use them in audio classification tasks. We are going to present every step of the descriptor extraction scheme with emphasis on the descriptor calculation procedures. The basic signal processing step is the STFT ("signal frame" and "FFT").

Temporal Modeling

This step accounts for the estimation of the attack and release times as described in [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF]. Firstly we calculate the amplitude (or temporal) envelope, which is a smooth curve that outlines the waveform. We estimate the attack and release times from here. It is important to note that the energy envelope itself must be interpolated in the morphing process. The next steps of the descriptor calculation scheme are repeated for every signal frame, such that variations naturally arising from the (presumably) acoustical nature of the sound source will give rise to shimmer and jitter.

Spectral Shape

The calculation of the spectral shape descriptors consists of three steps, spectral envelope estimation, application of the perceptual model, and finally calculation of the spectral shape descriptors, namely, spectral centroid, spread, skewness, kurtosis and slope [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF]. For every frame, we calculate the spectral envelope using a cepstral smoothing technique (true envelope [START_REF] Villavicencio | Improving LPC Spectral Envelope Extraction of Voiced Speech by True Envelope Estimation[END_REF]). Next, we apply the perceptual model, which consists of the mid-ear filter shown on the right of Figure 3 evaluated on the mel scale. We should notice that the result is similar to the MFCCbased spectral envelope used in [START_REF] Slaney | Automatic Audio Morphing[END_REF] without critical band smoothing. Finally we calculate the spectral shape descriptors for the mid-ear attenuated, mel-warped spectral envelope.

Harmonic Modeling

Here, we need to finally extract the remaining pitch information, i.e., the instantaneous values of the frequencies of the partials. There are many possible ways to do this, but for the sake of fidelity, we chose to perform an SMS-based sinusoidal plus residual analysis [START_REF] Serra | Musical Sound Modeling with Sinusoids Plus Noise[END_REF] (again on every signal frame) and keep only the frequency values of the sinusoidal part. The amplitudes of the partials are already accounted for by the spectral envelope estimation step. Temporal variations on the frequencies of the partials guarantee the naturalness of the tone.

Noise Modeling

The result of the SMS analysis is a sinusoidal component and a residual that models the noise part of the sound signal. In order to account for this perceptually important feature, we extract the spectral envelope and repeat the 'spectral shape' analysis here. The residual is modeled as pink noise modulated by the envelopes frame by frame.

MORPHING BY DESCRIPTORS

The final step of the morphing process consists of morphing between the descriptors with a desired morphing factor α and then resynthesizing a sound with parameter values that correspond to the morphed features. Some temporal features are somewhat independent from the spectral ones (attack and release times), while others are intrinsically intertwined with them (jitter, shimmer), such that we can manipulate attack an release times by time stretch/compress completely independently from other features, but jitter and shimmer are intrinsically contained in the time-varying nature of the analysis and will naturally morph as we interpolate the parameters. Our approach relies on the alignment of temporal features such as attack and release time, and a spectral envelope morphing technique that produces intermediate envelopes with the desired form (number of peaks) and intermediate spectral shape features. The tricky part is exactly the mapping between spectral shape descriptors and spectral envelope parameters. As other authors noted earlier [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF], [START_REF] Caetano | Evolutionary Spectral Envelope Morphing by Spectral Shape Descriptors[END_REF], [START_REF] Hikichi | Sound Timbre Interpolation Based on Physical Modeling[END_REF], linear variation of most spectral envelope parameters does not guarantee that the perceptual features will also change linearly, so we will present a study about which spectral envelope representations closely approximate linear interpolations in the descriptor space when linearly interpolated. Ezzat [START_REF] Ezzat | Morphing Spectral Envelopes using Audio Flow[END_REF] briefly reviews techniques to morph spectral envelopes. First they acknowledge that simply interpolating the envelope curve does not account for proper formant shifting. We should mention that this is exactly what most techniques do when they directly interpolate the amplitudes of a sinusoidal model. Then, they state that interpolating alternative representations of the envelopes, such as linear prediction or cepstral parameters, also poses problems and propose to use dynamic frequency warping (DFW) instead. So, the main motivation of this section is to verify this claim by investigating the perceptual impact of several spectral envelope interpolation schemes [START_REF] Paliwal | Interpolation Properties of Linear Prediction Parametric Representations[END_REF], namely, the envelope curve (ENV), linear prediction coefficients (LPC), reflection coefficients (RC), line spectral frequencies (LSF), cepstral coefficients (CC) and dynamic frequency warping (DFW). The rest of this section explains each step in our morphing technique.

Temporal Alignment

First, using the end of attack and beginning of release times estimated [START_REF] Peeters | A large set of audio features for sound description (similarity and classification) in the CUIDADO project[END_REF], we time stretch or compress the attack, sustain and release portions of both sounds to align them temporally [START_REF] Tellman | Timbre Morphing of Sounds with Unequal Numbers of Features[END_REF], [START_REF] Handel | Timbre perception and auditory object identification[END_REF]. For the attack and release times we use logarithmic interpolation.

Spectral Envelope Shape

We represent morphing by descriptors as weighted interpolation in the feature space representation, much like in [START_REF] Ahmad | Morphing of Transient Sounds Based on Shift-Invariant Wavelet Transform and Singular Value Decomposition[END_REF], [START_REF] Slaney | Automatic Audio Morphing[END_REF]. The fundamental difference is that our space corresponds to perceptual dimensions so there is no direct inversion for resynthesis. Instead, we are trying to find the spectral envelope model whose associated descriptors interpolate the closest possible to linearly when its parameters are linearly interpolated. Figure 5 the impact on the spectral descriptor domain interpolating cepstral, linear prediction, and dynamic frequency warping based spectral envelope model parameters for very challenging envelopes. On the Figure 5 shows the source and in solid lines and nine envelopes corresponding to linearly varying the interpolation factor by 0.1 steps dashed and dotted lines; on the right, we see the associated values of the spectral shape descriptors for each step. When evaluating Figure 5 we have to take into account spectral from and shape, that is, we want the envelope model that accounts properly for formant shifting and whose spectral shape descriptors vary as a straight line. The apparent difference in shape of the source and target for linear prediction based envelopes (LPC, RC and LSF) is due to the conversion from cepstral estimation. The conversion from cepstral to linear prediction based spectral envelope introduces artifacts, but we still consider that the result is better than extracting the envelope directly with linear prediction [START_REF] Villavicencio | Improving LPC Spectral Envelope Extraction of Voiced Speech by True Envelope Estimation[END_REF]. Figure 5 confirms for this case (we will extrapolate the conclusions) that interpolating envelope curves does not account for formant shifting and most spectral shape descriptors do not vary in a straight line. Moorer [START_REF] Moorer | The Use of Linear Prediction of Speech in Computer Music Applications[END_REF] states that LPCs do not interpolate well because they are derived from impulse responses, and therefore too sensitive to changes, and Figure 5 seems to confirm that. Figure 5 also shows that the linear interpolation of cepstral based envelope representations like Slaney [START_REF] Slaney | Automatic Audio Morphing[END_REF] proposes neither shifts the formants nor results in linear variation of descriptors. The same applies for the DFW based spectral envelope morphing proposed by Ezzat [START_REF] Ezzat | Morphing Spectral Envelopes using Audio Flow[END_REF]. On the other hand, RC and LSF behave fairly well under both constraints in this case just like Paliwal [START_REF] Paliwal | Interpolation Properties of Linear Prediction Parametric Representations[END_REF] states for LSFs. The only inconvenient could be the initial distortion caused by the conversion from using a cepstral smoothing envelope estimation technique.

Harmonic structure

Here we propose to morph quasi-harmonic musical instrument sounds with the same pitch, so that the partials have a one to one correspondence and no pitch shift is required. Since the spectral shape and form are morphed separately with the spectral envelope, we simply interpolate the partials frequency values to account for frequency fluctuations (jitter, shimmer), inharmonicity and other temporal features that are encoded in the frequency variation with time.

Amplitude Envelope

Here we simply interpolate the amplitude envelope curve and modulate the amplitude of the morphed sinusoidal component with it.

Stochastic Residual

We morph the spectral envelopes of the residual noise signal and synthesize a morphed residual by filtering pink noise with it and mixing it into the morphed sinusoidal component.

CONCLUSIONS AND FUTURE PERSPECTIVES

In this work, we describe techniques to automatically morph salient timbral dimensions of quasi-harmonic musical instrument sounds guided by high-level descriptors. High-level descriptors are acoustic correlates of timbre dimensions obtained in psychoacoustic studies, such that sounds whose features are intermediate between two would be placed between them in the underlying timbre space. So, interpolating the descriptor values becomes the goal itself to render the results more perceptually meaningful. We also reviewed the definitions and goals of sound morphing in the literature to try and establish common grounds for future research. Moreover, we reviewed the morphing techniques proposed so far and whether they took the perceptual impact into account. Finally, we evaluated the perceptual impact of interpolating the parameters of several spectral envelope models aiming to find which models correspond the closest to morphing in the underlying timbre space, that is, in the perceptual domain as measured by the descriptors. We investigated direct interpolation of the envelope curve, LPC, RC, LSF, CC, and DFW. We concluded that RC and LSF correspond the closest to morphing the descriptors linearly when linearly interpolated. Examples avaliable on http://recherche.ircam.fr/equipes/analysesynthese/caetano/icmc2010.html.

Future perspectives of this work include experimenting with different trajectories in timbre space determined by different time-varying morphing factors. It is also interesting to explore techniques to independently morph each timbre dimension by manipulating the descriptors with different morphing factors. Some technical aspects could be improved, such as extracting the temporal envelope for each partial and estimating the attacks independently to simulate onset asynchrony, include vibrato modeling and treatment, extending the technique to inharmonic sounds (would need different interpolation of harmonic structure), improving the estimation of attack time for percussive or plucked sounds. Also tremolo could be dealt with by developing a better energy envelope morphing than simply interpolate the curves. Finally, we could possibly extend the model to any sound object to finally be able to obtain a 'barking trumpet', for example.

ACKNOWDGEMENTS

This work is supported by the Brazilian Governmental Research Agency CAPES (process 4082-05-2).

Figure 1 .

 1 Figure 1. Depiction of image morphing to exemplify the aim of sound morphing.

Figure 2 .

 2 Figure 2. Depiction of two movies shown frame by frame.

Figure 3 .

 3 Figure 3. Left: Illustration of two-dimensional timbre space with two sound objects depicted as the circle and the square and one intermediate sound object depicted as the square with rounded corners. Right: Mid-ear filter applied to the spectral envelopes.

Figure 4 .

 4 Figure 4. Simplified scheme to calculate the descriptors.

Figure 5 .

 5 Figure 5. Perceptual impact of interpolating between the parameters of several spectral envelope models. The curves are shown on the left and the corresponding descriptor variation on the right.


Lingmin Liao
email: lingmin.liao@u-pec.fr

Wolfgang Steiner
email: steiner@liafa.jussieu.fr

DYNAMICAL PROPERTIES OF THE NEGATIVE BETA TRANSFORMATION

We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the negative beta transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (-β)-transformation is exact for all β > 1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (-β)-expansion of 1 when β tends to 1 is related to the Thue-Morse sequence.

A consequence of the exactness is that every Yrrap number, which is a β > 1 such that the (-β)-expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

INTRODUCTION

For a real number β > 1, the β-transformation is defined by T β : [0, 1) → [0, 1) , x → βx -⌊βx⌋ .

Rényi [START_REF] Rényi | Representations for real numbers and their ergodic properties[END_REF] was the first to use it for representing real numbers in base β, generalising expansions in integer bases. The (greedy) β-expansion of x ∈ [0, 1) is

x = ⌊βx⌋ β + ⌊β T β (x)⌋ β 2 + ⌊β T 2 β (x)⌋ β 3 + • • • .
In this paper, we study the (-β)-transformation (negative beta transformation)

T -β : (0, 1] → (0, 1] , x → -βx + ⌊βx⌋ + 1 .

Note that T -β (x) = -βx -⌊-βx⌋ except for finitely many points, hence T -β is a natural modification of the β-transformation, avoiding the discontinuity of x → -βx -⌊-βx⌋ at 0. For x ∈ (0, 1], set

d -β (x) = d -β,1 (x) d -β,2 (x) • • • with d -β,1 (x) = ⌊βx⌋ + 1 , d -β,n (x) = d -β,1 (T n-1 -β (x)) for n ≥ 1 , then the (-β)-expansion of x is x = ∞ k=1 -d -β,k (x) (-β) k = d -β,1 (x) β - d -β,2 (x) β 2 + d -β,3 (x) β 3 - d -β,4 (x) β 4 + • • • .
Examples of a β-transformation and a (-β)-transformation are depicted in Figure 1. The map T -β has been studied recently by Góra [START_REF] Góra | Invariant densities for generalized β-maps[END_REF] and Faller [START_REF] Faller | Contribution to the ergodic theory of piecewise monotone continuous maps[END_REF]. Ito and Sadahiro [START_REF] Ito | Beta-expansions with negative bases[END_REF] considered a (-β)-transformation on the interval -β β+1 , 1 β+1 , defined by x → -βx -β β+1βx . We remark that their transformation is conjugate to our T -β through the conjugacy function φ(x) = 1 β+1x. So, all the results in [START_REF] Ito | Beta-expansions with negative bases[END_REF] can be translated into our case directly, in particular their (-β)-expansion of φ(x)

∈ -β β+1 , 1 β+1 is φ(x) = ∞ k=1 d -β,k (x) -1 (-β) k .
Both β-transformation and (-β)-transformation are examples of piecewise monotonic transformations. Dynamical properties of general piecewise monotonic transformations have been investigated in the literature [START_REF]On intrinsic ergodicity of piecewise monotonic transformations with positive entropy[END_REF][START_REF]On intrinsic ergodicity of piecewise monotonic transformations with positive entropy[END_REF][START_REF]The structure of piecewise monotonic transformations[END_REF][START_REF] Keller | Piecewise monotonic transformations and exactness[END_REF][START_REF] Wagner | The ergodic behaviour of piecewise monotonic transformations[END_REF]. However, for the specific case of the (-β)-transformation, the detailed dynamical properties are unknown. As pointed out in [START_REF] Ito | Beta-expansions with negative bases[END_REF], for any β > 1, T -β admits a unique absolutely continuous invariant measure (hence ergodic), with the density

h -β (x) = n≥1, T n -β (1)≥x 1 (-β) n .
Contrary to the classical β-transformation, Ito and Sadahiro [START_REF] Ito | Beta-expansions with negative bases[END_REF] gave an example such that the density h -β is zero on some intervals.

In the present paper, we fully study this phenomenon. An interval on which the density equals zero will be called a gap. From a result of Keller [START_REF] Keller | Piecewise monotonic transformations and exactness[END_REF], one can deduce that, for fixed β > 1, the number of gaps is finite. In Theorem 2.1, we describe the set of the gaps and show that, as β decreases, the number of gaps forms an increasing sequence

0 < 1 < 2 < 5 < 10 < • • • < ⌊2 n+1 /3⌋ < • • • .
Figures 2 and3 show examples with 2 gaps and 5 gaps respectively. The endpoints of the gaps are determined by the orbit of 1, which is described by the sequence d -β [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF].

We also show (Theorem 2.5) that

lim β→1 d -β (1) = 211222112112112221122 • • • ,
which is the fixed point of the following morphism:

ϕ : 1 → 2 , 2 → 211 .
Here, ϕ is a morphism on finite words on the alphabet {1, 2}, which is naturally extended to the set of infinite words {1, 2} N as in [START_REF] Lothaire | Combinatorics on words[END_REF], and lim β→1 d -β (1) = u means that longer and longer prefixes of d -β (1) agree with prefixes of u when β → 1. We remark that this sequence has been studied e.g. in [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF][START_REF] Dubickas | On the distance from a rational power to the nearest integer[END_REF][START_REF]On a sequence related to that of Thue-Morse and its applications[END_REF], and is known to be the smallest aperiodic sequence in {1, 2} N with the property that all its proper suffixes are smaller than itself with respect to the alternate order. Moreover, by adding a 1 at the beginning, we obtain the sequence of run lengths in the Thue-Morse sequence. It is shown in [START_REF] Ito | Beta-expansions with negative bases[END_REF] that every sequence d -β (1), β > 1, has the property that all its proper suffixes are smaller than itself with respect to the alternate order.

A dynamical system (X, T ) is said to be locally eventually onto or topologically exact if, for any non-empty open subset U ⊂ X, there exists a positive integer n such that f n (U ) = X. The main theorem of this paper (Theorem 2.2) asserts that T -β is locally eventually onto on (0, 1] \ G(β), where G(β) is the (finite) union of the gaps.

The notion of topological exactness is derived from the exactness in ergodic theory introduced by Rohlin [START_REF] Rohlin | Exact endomorphisms of a Lebesgue space[END_REF]. A measure-preserving transformation T on a probability space (X, B, µ) is exact if and only if, for any positive measure subset A with T n (A) ∈ B (n ≥ 0), one has lim n→∞ µ(T n (A)) = 1 (see [18, p. 125]). As a corollary of our main theorem, we confirm Góra's conjecture [START_REF] Góra | Invariant densities for generalized β-maps[END_REF] that all (-β)-transformations are exact with respect to the unique absolutely continuous invariant measure.

An invariant measure of a dynamical system is called a maximal entropy measure if it maximises entropy. If there is a unique maximal entropy measure, the dynamical system is called intrinsic. As another application of our main theorem, we give a complete answer about the uniqueness of maximal entropy measures discussed in the thesis of Faller [START_REF] Faller | Contribution to the ergodic theory of piecewise monotone continuous maps[END_REF]. We prove that all (-β)-transformations are intrinsic (Corollary 2.4).

Finally, we use the main result to show that every number β > 1 with eventually periodic T -β -orbit of 1 is a Perron number. Recall that a number β > 1 with the corresponding property for T β is called a Parry number in reference to [START_REF] Parry | On the β-expansions of real numbers[END_REF], and note that T -β (x) = 1 -T β (x) for all x ∈ (0, 1). Because of this orientation reversing property, we call a number β > 1 with eventually periodic T -β -orbit of 1 an Yrrap number. Parry numbers are also known to be Perron numbers. However, not every Perron number is a Parry number or an Yrrap number. In Section 6, we give examples showing that the set of Parry numbers and the set of Yrrap numbers do not include each other.

The paper is organised as follows. In Section 2, we state the main results. In Section 3, we establish properties of the sequences d -β [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF], which allow describing the structure of the gaps in Section 4. In Section 5, we show that T -β is locally eventually onto on (0, 1] \ G(β). The proofs of the main results are completed in Section 6.

We remark that, independently of our work, some of our results have been proved for slightly more general transformations by Hofbauer [START_REF] Hofbauer | Beta-transformations with negative slope[END_REF].

MAIN RESULTS

For each n ≥ 1, let γ n be the (unique) positive real number defined by

γ gn+1 n = γ n + 1 , with g n = ⌊2 n+1 /3⌋ ,
and set γ 0 = ∞. Then 2 > γ 1 > γ 2 > γ 3 > • • • > 1 .
Note that γ 1 is the golden ratio and that γ 2 is the smallest Pisot number.

For each n ≥ 0 and 1 < β < γ n , set

G n (β) = G m,k (β) | 0 ≤ m < n, 0 ≤ k < 2 m+1 +(-1) m 3 ,
with open intervals

G m,k (β) =    T 2 m+1 +k -β (1), T (2 m+2 -(-1) m )/3+k -β (1) if k is even, T (2 m+2 -(-1) m )/3+k -β (1), T 2 m+1 +k -β (1) if k is odd,
in particular G 0 (β) is the empty set.

Theorem 2.1. For any γ n+1 ≤ β < γ n , n ≥ 0, the set of gaps of the transformation

T -β is G n (β), which consists of g n = ⌊2 n+1 /3⌋ disjoint non-empty intervals.
We define therefore Góra [START_REF] Góra | Invariant densities for generalized β-maps[END_REF] proved that, for β > γ 2 , the transformation T -β is exact, and he conjectured that this would hold for all β > 1. We confirm his conjecture.

G(β) = I∈Gn(β) I if γ n+1 ≤ β < γ n , n ≥ 0 .
Corollary 2.3. For any β > 1, the transformation T -β is exact with respect to its unique absolutely continuous invariant measure.

Proof. The unicity of the absolutely continuous invariant measure follows from [START_REF] Li | Ergodic transformations from an interval into itself[END_REF], as already noted in [START_REF] Ito | Beta-expansions with negative bases[END_REF]. Now, the exactness is a direct consequence of Theorem 2.2.

Faller [START_REF] Faller | Contribution to the ergodic theory of piecewise monotone continuous maps[END_REF] The key to understanding the (-β)-transformation is to know the (-β)-expansion of 1. Here, the main tool will be the morphism ϕ : 1 → 2, 2 → 211. The expansion of 1 under T -γn can be described by this morphism, and the expansion of 1 under T -β tends to the fixed point of the morphism when β tends to 1 (from above). Theorem 2.5. For every n ≥ 1, we have

d -γn (1) = ϕ n-1 2 1 , where 1 = 111 • • • . If 1 < β ≤ γ n , then d -β (1) starts with ϕ n (2), hence lim β→1 d -β (1) = lim n→∞ ϕ n 2 1 = 211222112112112221122 • • • .
Masáková and Pelantová [START_REF] Masáková | Ito-Sadahiro numbers vs[END_REF] showed that the Yrrap numbers (called Ito-Sadahiro numbers by them) are algebraic integers with all conjugates having modulus less than 2, thus all Yrrap numbers β ≥ 2 are Perron numbers. Again, we give a complete answer.

Theorem 2.6. Every Yrrap number is a Perron number.

At the end of this section, we recall some identities. Note that

|ϕ n (1)| = 2 n+1 + (-1) n 3 = g n + 1 + (-1) n 2 , (2.1) |ϕ n (2)| = |ϕ n+1 (1)| = 2 n+2 -(-1) n 3 = g n+1 + 1 -(-1) n 2 , (2.2) |ϕ n (11)| = 2 • |ϕ n (1)| = 2 n+2 + 2 • (-1) n 3 = g n+1 + 1 + (-1) n 2 , (2.
G n (β) = G m,k (β) | 0 ≤ m < n, 0 ≤ k < |ϕ m (1)| , G m,k (β) =    T |ϕ m (21)|+k -β (1), T |ϕ m (2)|+k -β (1) if k is even, T |ϕ m (2)|+k -β (1), T |ϕ m (21)|+k -β (1) if k is odd.

POLYNOMIALS AND EXPANSIONS

To study the trajectories of 1 under T -β , we define the maps

f β,a : R → R , x → -βx + a , a ∈ {1, 2} .
The composition of maps f β,a1 , . . . , f β,a k , is denoted by

f β,a1•••a k = f β,a k • • • • • f β,a1 . Since f β,a1•••a k (1) = (-β) k + k j=1 a k (-β) k-j ,
we define the polynomial

P a1•••a k = (-X) k + k j=1 a k (-X) k-j ∈ Z[X] for every word a 1 • • • a k ∈ {1, 2} k , k ≥ 1. Then f β,a1•••a k (1) = P a1•••a k (β), thus T k -β (1) = f β,d -β,1 (1)•••d -β,k (1) (1) = P d -β,1 (1)•••d -β,k (1)
(β) . For the proof of Theorem 2.5, we use the following polynomial identities. Lemma 3.1. For 1 ≤ j < k, we have

P a1•••a k = (-X) k-j P a1•••aj -1 + P aj+1•••a k .
Proof. We can deduce the identity directly from the definition. Lemma 3.2. For every n ≥ 0, we have

X 1+(-1) n 2 P ϕ n (2) + X 1-(-1) n 2 P ϕ n (11) = X + 1 = X 1+(-1) n 2 + X 1-(-1) n 2 .
Proof. The second equation holds for even and odd n, thus for all n ∈ Z. Since

XP 2 + P 11 = X(-X + 2) + X 2 -X + 1 ,
the first equation holds for n = 0. For every n ≥ 0, we have

X 1+(-1) n+1 2 P ϕ n+1 (2) + X 1-(-1) n+1 2 P ϕ n+1 (11) = X 1-(-1) n 2 P ϕ n (211) + X 1+(-1) n 2 P ϕ n (22) .
Then, using that |ϕ n (11)| is even, |ϕ n (2)| is odd and Lemma 3.1, we obtain

X 1-(-1) n 2 P ϕ n (211) + X 1+(-1) n 2 P ϕ n (22) = X 1-(-1) n 2 +|ϕ n (11)| P ϕ n (2) -1 + X 1-(-1) n 2 P ϕ n (11) -X 1+(-1) n 2 +|ϕ n (2)| P ϕ n (2) -1 + X 1+(-1) n 2 P ϕ n (2) = X 1+(-1) n 2 P ϕ n (2) + X 1-(-1) n 2 P ϕ n (11) .
Therefore, the lemma follows inductively.

The following lemma is a consequence of Lemma 3.2.

Lemma 3.3. For every n ≥ 0, the words ϕ n (2) and ϕ n (11) agree on the first g n+1 -1 letters and differ on the g n+1 -st letter.

Proof. The polynomials X 1+(-1) n 2 P ϕ n (2) and X

1-(-1) n 2 P ϕ n (11) have degree g n+1 + 1 by (2.2) and (2.3). Thus the coefficient of X gn+1+1-j in their sum is equal to the difference of the j-th letter in ϕ n (2) and ϕ n (11), for 1 ≤ j ≤ min(|ϕ n (2)|, |ϕ n (11)|) = g n+1 . Since the sum of these polynomials is equal to X + 1, by Lemma 3.2, the first g n+1 -1 letters in ϕ n (2) and ϕ n (11) are equal, and the g n+1 -st letters differ. Lemma 3.4. For every n ≥ 0, we have

1 -P ϕ n (1) = X 1+(-1) n 2 n-1 m=0 X |ϕ m (1)| -1 , (3.1) P ϕ n (21) -1 = X gn+1+1 -X gn+1 -X 1+(-1) n 2 n-1 m=0 X |ϕ m (1)| -1 , (3.2) P ϕ n (21) -P ϕ n (2) = X gn+1+1 -X -1 n-1 m=0 X |ϕ m (1)| -1 , (3.3) P ϕ n (2) -P ϕ n+1 (2) = X gn+1+1 -X -1 n m=0 X |ϕ m (1)| -1 . (3.4)
Proof. Since 1 -P 1 = X, (3.1) holds for n = 0. By Lemmas 3.2 and 3.1, we obtain that

1 -P ϕ n+1 (1) = 1 -P ϕ n (2) = X -(-1) n P ϕ n (11) -1 = X -(-1) n -X |ϕ n (1)| P ϕ n (1) -1 + P ϕ n (1) -1 = X -(-1) n X |ϕ n (1)| -1 1 -P ϕ n (1)
for every n ≥ 0, hence (3.1) holds inductively for all n ≥ 0.

Applying first Lemma 3.1, then equation (3.1) for n and n + 1, we obtain that

P ϕ n (21) -1 = -X |ϕ n (1)| P ϕ n (2) -1 + P ϕ n (1) -1 = X |ϕ n (1)|+ 1-(-1) n 2 X |ϕ n (1)| -1 -X 1+(-1) n 2 n-1 m=0 X |ϕ m (1)| -1 .
By (2.1) and (2.3), we have With the help of Lemma 3.4, we will show the following proposition, where η n > 1, n ≥ 1, is defined by

|ϕ n (1)| + 1-(-1) n 2 = g n + 1 and 2 • |ϕ n (1)| + 1-(-1) n 2 = g n+1 + 1,
η gn+1 n = η gn-1+1 n + η 1-(-1) n 2 n .
It can be easily verified that

2 = η 1 > γ 2 2 > γ 1 > η 2 > γ 2 > η 3 > • • • > 1 . Proposition 3.5. Let β > 1, n ≥ 1. Then d -β (1) starts with ϕ n (2) if and only if β < γ 2 2 in case n = 1, β ≤ η n in case n is even, β < η n in case n ≥ 3 is odd.
For the proof of the proposition, we use the following lemmas. For the sake of readability, we often omit the dependence on β in T -β , d -β and f β,a in the sequel. Lemma 3.6. Let x ∈ (0, 1] and 1 < β < 2. Then

• d 1 (x) = 2 if and only if f 2 (x) ≤ 1, • d 1 (x) = 1 if and only if f 1 (x) > 0, • d 1 (x) d 2 (x) = 11 if and only if 0 < f 11 (x) < 1. Proof. For a ∈ {1, 2}, we have d 1 (x) = a if and only if f a (x) ∈ (0, 1]. The inequalities f 2 (x) > -β + 2 > 0 and f 1 (x) < 1 prove
the first two points. Noting that f 1 (x) < 1 is equivalent to x > 0, the third point follows from the second point.

Lemma 3.7. Let β > 1. If d(1) starts with a 1 • • • a k and d(f a1•••a k (1)) starts with b 1 • • • b j , then d(1) starts with a 1 • • • a k b 1 • • • b j . Proof. If d 1 (1) • • • d k (1) = a 1 • • • a k , then T k (1) = f a1•••a k (1)
, and the lemma follows from d k+1 (1)

• • • d k+j (1) = d 1 (T k (1)) • • • d j (T k (1)).
Lemma 3.8. Let x ∈ (0, 1], n ≥ 1, such that d(x) starts with the first g n -1 letters of ϕ n (1). Then d(x) starts with ϕ n (1) if and only if

f ϕ n (1) (x) ≤ 1 in case n is odd, 0 < f ϕ n (1) (x) < 1 in case n is even.
Proof. Assume that d(x) starts with the first g n -1 letters of ϕ n (1), n ≥ 1. If n is odd, then g n = |ϕ n (1)|, and ϕ n (1) ends with 2, thus f 2 T |ϕ n (1)|-1 (1) = f ϕ n (1) [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF]. By Lemma 3.6, we have

d |ϕ n (1)| (1) = d 1 (T |ϕ n (1)|-1 (1)) = 2 if and only if f 2 T |ϕ n (1)|-1 (1) ≤ 1. If n is even, then g n = |ϕ n (1)| -1, and ϕ n (1) ends with 11, thus f 11 T |ϕ n (1)|-2 (1) = f ϕ n (1) (1). By Lemma 3.6, d |ϕ n (1)|-1 (1) d |ϕ n (1)| (1) = 11 is equivalent to 0 < f 11 T |ϕ n (1)|-2 (1) < 1. Lemma 3.9. Let β > 1, n ≥ 1. If d(1) starts with ϕ n (2), then T gn-1 is continuous on T |ϕ n (1)| (1), 1 , and 1/β is in the interior of T gn-1 T |ϕ n (1)| (1), 1 .
Proof. Let β > 1 and n ≥ 1 such that d(1) starts with ϕ n (2) = ϕ n-1 (211), then d(T |ϕ n (1)| (1)) starts with ϕ n-1 [START_REF]The structure of piecewise monotonic transformations[END_REF]. By Lemma 3.3, the words ϕ n-1 (2) and ϕ n-1 (11) share the first g n -1 letters and differ on the g n -th letter. This proves that T gn-1 is continuous on T |ϕ n (1)| (1), 1 , and that 1/β is in the interior or the right endpoint of

T gn-1 T |ϕ n (1)| (1), 1 . If T gn-1 (1) = 1/β with odd n, then T |ϕ n (1)| (1) = 1. If T |ϕ n (1)|+gn-1 (1) = T gn+1-1 (1) = 1/β with even n, then T |ϕ n (2)| (1) = 1. Both situations are impossible since f ϕ n (1) (1) < 1 for all β > 1, n ≥ 0, by (3.1). Therefore, 1/β is in the interior of T gn-1 T |ϕ n (1)| (1), 1 . Lemma 3.10. For any 1 < β ≤ η n , n ≥ 1, we have f ϕ n-1 (1) (1) ≤ f ϕ n (2) (1) < 1.
Proof. By (3.1), we have f ϕ n (1) (1) < 1 for all β > 1, n ≥ 0.

For all 1 < β ≤ η n , n ≥ 1, we have f ϕ n-1 (21) (1) ≤ 1 by (3.2). Since f ϕ n-1 (1) is order reversing, we get that

f ϕ n (2) (1) = f ϕ n-1 (211) (1) ≥ f ϕ n-1 (1) (1).
Proof of Proposition 3.5. Let β > 1. First note that d 1 (1) = 2 if and only if β < 2. Then, by Lemma 3.6, d(1) starts with ϕ(2) if and only if 0 < f ϕ(2) (1) < 1. By (3.1), we have

1 -f ϕ(2) (1) = β(β -1) 2 > 0, and 1 -f ϕ(2) (1) < 1 if and only if β -1 < β -1/2 , i.e., β < γ 2
2 . This proves the case n = 1 of the proposition. Suppose now that d(1) starts with ϕ n+1 (2) = ϕ n (211) for some n ≥ 1. Then we must have f ϕ n (21) (1) ≤ 1, which is equivalent to β ≤ η n+1 by (3.2). Suppose moreover that the proposition is already shown for n. Since η n+1 < min(η n , γ 2 2 ), this implies that d(1) starts with ϕ n (2). Furthermore, η n+1 < γ n and (3.4) yield that [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF]. By Lemma 3.9, the first g n -1 letters of d(x) are equal to those of d( 1) for all x ∈ f ϕ n (1) (1), 1 . Therefore, Lemma 3.8 implies that d(1) starts with ϕ n (21) if and only if f ϕ n (21) (1) ≤ 1 in case n is odd, 0 < f ϕ n (21) (1) < 1 in case n is even. For even n, we must therefore have β < η n+1 . Note that f ϕ n (21) (1) > 0 is guaranteed by

f ϕ n (2) (1) > f ϕ n (1)
f ϕ n (21) (1) > f ϕ n (1) (1) = T |ϕ n (1)| (1)
, which follows from f ϕ n (2) (1) < 1 and the order reversing property of f ϕ n (1) . By Lemma 3.10, we have

f ϕ n (1) (1) ≤ f ϕ n (211) (1) < 1, thus f ϕ n (21) (1) ∈ f ϕ n (1) (1), 1 , which implies that d(f ϕ n (21) (1)
) starts with ϕ n (1). Putting everything together, we obtain that d(1) starts with ϕ n (211) = ϕ n+1 (2) if and only if β < η n+1 with even n, or β ≤ η n+1 with odd n.

We conclude the section with the proof of Theorem 2.5.

Proof of Theorem 2.5. By Proposition 3.5, d -β (1) starts with ϕ n (2) for any 1 < β ≤ γ n , n ≥ 1. By (3.3), we have f γn,ϕ n-1 (21

) (1) = f γn,ϕ n-1 (2) (1), thus T |ϕ n-1 (21)| -γn (1) = T |ϕ n-1 (2)| -γn (1), i.e., T |ϕ n-1 (2)| -γn (1) is a fixed point of T |ϕ n-1 (1)| -γn
. Hence, by Lemma 3.7, d -γn (1) = ϕ n-1 (2 1). Since lim n→∞ γ n = 1, we obtain that d -β (1) starts with ϕ n (2) for larger and larger n when β → 1, thus lim β→1 d -β (1) = lim n→∞ ϕ n (2 1).

STRUCTURE OF THE GAPS

We will show that the support of the invariant measure is

F (β) = (0, 1] \ G(β) . For β > 1, n ≥ 0, let F n (β) = F n,k (β) | 0 ≤ k ≤ g n with F n,k (β) =    T |ϕ n (1)|+k -β (1), T k -β (1) if k is even, k < g n , T k -β (1), T |ϕ n (1)|+k -β (1) if k is odd, k < g n , F n,gn (β) =    0, T |ϕ n (1)|-1 -β (1) if n is even, 0, T |ϕ n (2)|-1 -β (1) if n is odd.
(We will consider these sets only in the case when the left number is smaller than the right number.) Figures 2 and3 show examples of the decomposition of (0, 1] into these sets.

As in Section 3, we often omit the dependence on β in the following.

Lemma 4.1. Let n ≥ 2 and 1 < β ≤ η n if n is even, 1 < β < η n if n is odd; or let n = 1, 1 < β < γ 2 2 . Then we have (i) F n,k = T k (F n,0 ) for all 0 ≤ k < g n , (ii) 1/β is in the interior of F n,gn-1 , (iii) T gn (F n,0 ) = F n,gn ∪ F n,0 , T (F n,gn ) = F n+1,0 \ {1}, if n is odd, (iv) T gn (F n,0 ) = F n,gn ∪ F n+1,0 , T (F n,gn ) = F n,0 \ {1}, if n is even.
Proof. Let β and n have the properties of the statement of the lemma. Then Proposition 3.5 and Lemma 3.9 give the points i. and ii.

If n is odd, then F n,gn-1 = T gn+1-1 (1), T gn-1 (1) , thus

T (F n,gn-1 ) = 0, T gn+1 (1) ∪ T gn (1), 1 = 0, T |ϕ n (2)|-1 (1) ∪ T |ϕ n (1)| (1), 1 = F n,gn ∪ F n,0 .
Since d(1) starts with ϕ n (2) and ϕ n (2) ends with 1, we have

T (F n,gn ) = T 0, T |ϕ n (2)|-1 (1) = T |ϕ n (2)| (1), 1 = F n+1,0 \ {1} ,
i.e., iii. holds. If n is even, then F n,gn-1 = T gn-1 (1), T gn+1-1 (1) and

T (F n,gn-1 ) = 0, T gn (1) ∪ T gn+1 (1), 1 = 0, T |ϕ n (1)|-1 (1) ∪ T |ϕ n (2)| (1), 1 = F n,gn ∪ F n+1,0 .
(i) F n-1,k = F n,|ϕ n-1 (1)|+k ∪ G n-1,k ∪ F n,k for all 0 ≤ k < |ϕ n-1 (1)|, (ii) G m,k = T k (G m,0 ) for all 0 ≤ m < n, 0 ≤ k < |ϕ m (1)|, (iii) T |ϕ m (1)| (G m,0 ) = G m,0 ∪ F m,0 \ F m+1,0 ⊃ G m,0 for all 0 ≤ m < n.
Proof. We have F 0 = {F 0,0 } = {(0, 1]} and G 0 = ∅, thus the elements of F 0 and G 0 form the trivial partition of (0, 1]. Let now 1 < β < γ n , n ≥ 1, and assume that F n-1 ∪ G n-1 forms a partition of (0, 1]. Lemma 3.4 and the order reversing property of f ϕ n-1 (1) give [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF]. Ratios between G 1,0 , G 2,0 and the rest of the picture are not respected. In reality, G 1,0 is almost 10 times the size of G 2,0 , and G 2,0 is almost 5 times the size of F 2,0 .

f ϕ n-1 (1) (1) < f ϕ n-1 (21) (1) < f ϕ n-1 (2) (1) < f ϕ n (2) (1) < 1 .
t k = T k -β (
For n = 1, we have f 21 (1) > 0. Therefore, F n-1,0 splits into F n,|ϕ n-1 (1)| , G n-1,0 and F n,0 , which are intervals of positive length. This shows i. in case n = 1. By Proposition 3.5 and Lemma 3.9, T gn-1-1 is continuous on F n-1,0 , thus every (21)| (1)) and d(1) start with ϕ n-1 (1). In this case,

F n-1,k , 0 ≤ k < g n-1 , also splits into F n,|ϕ n-1 (1)|+k , G n-1,k and F n,k . If n is even, then this proves i. since g n-1 = |ϕ n-1 (1)|. In this case, we have F n-1,gn-1 = F n,gn , thus F n ∪ G n is a refinement of the partition F n-1 ∪ G n-1 . If n is odd, then we use that T |ϕ n-1 (1)| is continuous on G n-1,0 ∪ F n-1,0 since both d(T |ϕ n-1
F n-1,gn-1 = F n,|ϕ n-1 (1)|-1 splits into F n,gn , G n-1,|ϕ n-1 (1)|-1 and F n,|ϕ n-1 (1)|-1 , thus i.
holds for odd n too, and

F n ∪ G n is again a refinement of the partition F n-1 ∪ G n-1 .
For all 0 ≤ m < n, the continuity of T |ϕ m (1)| on G m,0 gives ii. Moreover,

T |ϕ m (1)| (G m,0 ) = T |ϕ m (21)| (1), T |ϕ m+1 (2)| (1) and F m+1,0 ⊂ F m,0 imply iii.
Proof. If β ≥ γ 1 , then µ(G(β)) = µ(∅) = 0. We will show, by induction on n, that (4.1) µ(G n,k ) = 0 for all n ≥ 0 , 0 ≤ k < |ϕ n (1)| , 1 < β < γ n+1 . Since G = n-1 m=0 |ϕ m (1)|-1 k=0 G m,k for γ n+2 ≤ β < γ n+1 , this proves the proposition. For 1 < β < γ 1 , we have G 0,0 = T 2 (1), T (1) = f 1 T (1), T (1) and T -1 (G 0,0 ) = f -1 1 (G 0,0 ) = f -1 1 T (1), T (1) ⊂ G 0,0 . Since µ(T -1 (G 0,0 )) = µ(G 0,0 ), we obtain µ f 1 T (1), f -1 1 T (1) = 0. Iteratively, we get, for all k ≥ 0, µ f -2k+1 1 T (1), f -2k-1 1 T (1) = 0 , µ f -2k-2 1 T (1), f -2k 1 T (1) = 0 . This gives µ T 2 (1), 1 β+1 = 0 and µ 1 β+1 , T (1) = 0, since f -1
1 is contracting with fixed point 1 β+1 . Then, the absolute continuity of µ implies µ(G 0,0 ) = 0, hence (4.1) holds for n = 0. Now consider 1 < β < γ n+1 , n ≥ 1, and assume that (4.1) holds for n -1. By Proposition 4.2 i.-iii., we have (4.2)

T (G m,k-1 ) = G m,k if 1 ≤ k < |ϕ m (1)|, 0 ≤ m ≤ n, G m,0 ∪ F m+1,|ϕ m (1)| ∪ G m+1,0 if k = |ϕ m (1)|, 0 ≤ m < n. Since T -1 I∈Gn I ⊂ I∈Gn I by Corollary 4.3, we obtain that T -1 (G n,0 ) ⊂ G n,|ϕ n (1)|-1 ∪ G n-1,|ϕ n-1 (1)|-1 and T 1-|ϕ n (1)| (G n,|ϕ n (1)|-1 ) = G n,0 . Thus, up to a set of µ-measure zero, T -|ϕ n (1)| (G n,0 ) = f -1 ϕ n (1) (G n,0 ) ⊂ G n,0 , with f -1 ϕ n (1) (G n,0 ) = f -1 ϕ n (1) f ϕ n (1) T |ϕ n (2)| (1), T |ϕ n (2)| (1) = f -1 ϕ n (1) T |ϕ n (2)| (1), T |ϕ n (2)| (1)
. As in the case n = 0, using the T -invariance of µ, we obtain that

µ f -2k+1 ϕ n (1) T |ϕ n (2)| (1), f -2k-1 ϕ n (1) T |ϕ n (2)| (1) = 0 , µ f -2k-2 ϕ n (1) T |ϕ n (2)| (1), f -2k ϕ n (1) T |ϕ n (2)| (1) = 0 , for all k ≥ 0. Thus µ T |ϕ n (21)| (1), y = 0 and µ y, T |ϕ n (2)| (1) = 0, where y denotes the fixed point of f ϕ n (1) , i.e., µ(G n,0 ) = 0. Since µ(G n,k ) = µ(T -k (G n,k )) = µ(G n,0 ) for 1 ≤ k < |ϕ n (1)|, (4.1) holds for n. Proposition 4.5. Let β > 1, then lim k→∞ λ T -k -β (G(β)) = 0.
Proof. If β ≥ γ 1 , then G(β) = ∅ and the statement holds. For 1 < β < γ n+1 , n ≥ 0, the preimage T -k (G n,0 ) can be written as a disjoint union

T -k (G n,0 ) = a1•••a k ∈L k f -1 a1•••a k (G n,0 )
with a set of words

L k ⊆ {1, 2} k . Then we have µ(T -k (G n,0 )) = #(L k ) β -k µ(G n,0
µ(T -k (G n,0 )) = 0. As T -j (G n,j ) = G n,0 for 1 ≤ j < |ϕ n (1)|, we also have lim k→∞ µ(T -k (G n,j )) = 0, thus lim k→∞ µ(T -k (I)) = 0 for all I ∈ G n+1 .

LOCALLY EVENTUALLY ONTO

Lemma 5.1. Let 1 < β < γ 1 , I be an interval of positive length in F (β), and n ≥ 1 such that η n+1 < β < η n with odd n ≥ 1, or η n+1 ≤ β ≤ η n with even n ≥ 2. Then there exists an m ≥ 0 such that T m -β (I) ⊇ F n,0 (β) \ {1}.

Proof. We use ideas of Góra's proof of Proposition 8 in [START_REF] Góra | Invariant densities for generalized β-maps[END_REF]. We distinguish four cases.

Case 1: odd n ≥ 1, η n+1 < β < γ n . In this case, we have

β 2gn = β gn+1 > β gn + β -1 ≥ β + β -1 > 2.
Therefore, the largest connected components of images of I grow until they cover two consecutive discontinuities of T 2gn in the interior of F (β). Since the only discontinuity of T is 1/β ∈ F n,gn-1 , the images eventually also cover two consecutive discontinuities of T 2gn in the interior of F n,gn-1 = T |ϕ n (2)|-2 (1), T |ϕ n (1)|-1 (1) . We have

T 2gn 1/β, f -1 2ϕ n (1) (1) = T gn 1/β, T |ϕ n (1)|-1 (1) , T 2gn f -1 11ϕ n (1) (1), 1/β = T gn-1 1/β, T |ϕ n (1)|-1 (1) , T 2gn f -1 2ϕ n (1) (1), T |ϕ n (1)|-1 (1) = T gn T |ϕ n (2)|-2 (1), 1/β , T 2gn T |ϕ n (2)|-2 (1), f -1 11ϕ n (1) (1) = T gn-1 T |ϕ n (21)|-1 (1), 1/β .
Using Lemma 3.9, we obtain that T 2gn is continuous on these four intervals. Note that

T |ϕ n (2)|-2 (1) < f -1 11ϕ n (1) (1) since f 11ϕ n (1) T |ϕ n (2)|-2 (1) = f ϕ n (21) (1) > 1. The discontinuities of T 2gn in the interior of F n,gn-1 are therefore 1/β, f -1 2ϕ n (1) (1) and f -1 11ϕ n (1) (1). With T gn+1 1/β, f -1 2ϕ n (1) (1) = F n,0 and T gn+1 1/β, T |ϕ n (1)|-1 (1) = F n,0 \ {1} ,
we obtain that T m (I) ⊇ F n,0 \ {1} for some m ≥ 0.

Case 2: odd n ≥ 1, γ n+1 ≤ β ≤ η n+1 . Since β gn+1+1 ≥ β + 1 > 2, we look at T 2gn+1 on F n,gn-1 . Note that T 2gn 1/β, f -1 2ϕ n (1) (1) = F n,gn-1 , thus there is a discontinuity of T 2gn+1 in the interior of 1/β, f -1 2ϕ n (1) [START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF] . Contrary to Case 1, T 2gn+1 has no discontinuity in F n,gn-1 on the left of 1/β. More precisely,

T 2gn+1 1/β, f -1 2ϕ n (2) (1) = T T |ϕ n (2)|-2 (1), 1/β , T 2gn+1 f -1 2ϕ n (2) (1), f -1 2ϕ n (1) (1) = T 1/β, T |ϕ n (1)|-1 (1) , T 2gn+1 f -1 2ϕ n (1) (1), T |ϕ n (1)|-1 (1) = T gn+1 T |ϕ n (2)|-2 (1), 1/β , T 2gn+1 T |ϕ n (2)|-2 (1), 1/β = T gn T |ϕ n (21)|-1 (1), T |ϕ n (1)|-1 (1) .
Here,

f ϕ n (21) (1) ≤ 1 implies that T |ϕ n (21)|-1 (1) = f -1 2 f ϕ n (21) (1) ≥ 1/β. Hence, the discontinuities of T 2gn+1 in the interior of F n,gn-1 are 1/β, f -1 2ϕ n (2) (1) and f -1 2ϕ n (1) (1), with T 2gn+2 1/β, f -1 2ϕ n (2) (1) = F n+1,0 \ {1} , T 2gn+1 f -1 2ϕ n (2) (1), f -1 2ϕ n (1) (1) = F n,0 .
This gives T m (I) ⊇ F n+1,0 \ {1} for some m ≥ 0.

Case 3: even n ≥ 2, η n+1 ≤ β < γ n . In this case, we have

β 2gn = β gn+1-1 > β gn-1 + β -1 ≥ β + β -1 > 2, F n,gn-1 = T |ϕ n (1)|-2 (1), T |ϕ n (2)|-1 (1)
, and

T 2gn 1/β, f -1 2ϕ n (1) (1) = T gn T |ϕ n (1)|-2 (1), 1/β , T 2gn f -1 11ϕ n (1) (1), 1/β = T gn-1 T |ϕ n (1)|-2 (1), 1/β , T 2gn f -1 2ϕ n (1) (1), T |ϕ n (2)|-1 (1) = T gn 1/β, T |ϕ n (21)|-2 (1) , T 2gn T |ϕ n (1)|-2 (1), f -1 11ϕ n (1) (1) = T gn-1 1/β, T |ϕ n (2)|-1 (1) .
Here, Since β > η n+1 , we have T |ϕ n (21)|-1 (1) < 1/β, thus T gn (F n+1,0 ) ⊇ F n,0 , and

f 2ϕ n (1) T |ϕ n (2)|-1 (1) = f ϕ n (21) (1) ≥ 1 implies f -1 2ϕ n (1) (1) ≤ T |ϕ n (2)|-1 (1). The discontinuities of T 2gn in the interior of F n,gn-1 are at most 1/β, f -1 2ϕ n (1) (1) and f -1 11ϕ n (1) (1), with T gn+2 1/β, f -1 2ϕ n (1) (1) = F n,0 \ {1} , T gn+3 f -1 11ϕ n (1) (1), 1/β = F n,0 \ {1, T |ϕ n (1)| (1)} , similarly to Case 1. We obtain that T m (I) ⊇ F n,0 \ {1, T |ϕ n (1)| (1)} for some m ≥ 0. Since T gn+1 F n,0 \ {1, T |ϕ n (1)| (1)} ⊇ F n,0 \ {1}, we also have T m (I) ⊇ F n,0 \ {1} for some m ≥ 0. Case 4: even n ≥ 2, γ n+1 ≤ β < η n+1 . Since β gn+1+1 ≥ β + 1 > 2, we look at T 2gn+2 on F n,gn-1 . We have T 2gn+1 1/β, f -1 2ϕ n (1) (1) = F n,gn-1 \ {T |ϕ n (1)|-2 (1)} with f -1 2ϕ n (1) (1) > T |ϕ n (2)|-1 (1), thus f -1 2ϕ n (1) (1) is outside of F n,gn-1 , but T 2gn+2 still has a discontinuity in 1/β, T |ϕ n (2)|-1 (1) . More precisely, T 2gn+2 1/β, f -1 2ϕ n (2) (1) = T 1/β, T |ϕ n (2)|-1 (1) , T 2gn+2 f -1 11ϕ n (1) (1), 1/β = T gn+1 T |ϕ n (1)|-2 (1), 1/β , T 2gn+2 f -1 2ϕ n (2) (1), T |ϕ n (2)|-1 (1) = T T |ϕ n (2)|-2 (1), 1/β , T 2gn+2 T |ϕ n (1)|-2 (1), f -1 11ϕ n (1) (1) = T T |ϕ n (1)|-2 (1), T |ϕ n (21)|-2 (1) , with T |ϕ n (21)|-2 (1) = f -1 11 f ϕ n (21) (1) < 1/β since f ϕ n (21) (1) < 1. Therefore, the discontinuities of T 2gn+2 in the interior of F n,gn-1 are 1/β, f -1 2ϕ n (2) (1) and f -1 11ϕ n (1) (1), with T 2gn+3 1/β, f -1 2ϕ n (2) (1) = F n+1,0 , T gn+3 f -1 11ϕ n (1) (1), 1/β = F n,0 \ {1, T |ϕ n (1)| (1)} . This gives T m (I) ⊇ F n+1,0 \ {1} for some m ≥ 0. Lemma 5.2. Let 1 < β < γ 1 and n ≥ 1 such that η n+1 < β < η n with odd n ≥ 1, or η n+1 ≤ β ≤ η n with even n ≥ 2. Then we have T gn(2gn+(-1) n ) -β F n,0 (β) \ {1} ⊇ F (β) \ T k -β (1) | 0 ≤ k ≤ g n (2g n + (-1) n ) .
T 2gn+1 (F n,0 ) ⊇ F n,0 ∪ F n,1 .
Inductively, we get T (gn-1)(2gn+1) (F n,0 ) ⊇ gn-1 k=0 F n,k . Then, T (F n,gn-1 ) = F n,gn ∪ F n,0 yields that T gn(2gn-1) (F n,0 ) = T (gn-1)(2gn+1)+1 (F n,0 ) =

1

 1

FIGURE 1 .

 1 FIGURE 1. β-transformation (left) and (-β)-transformation (right), β = 1+ √ 5 2 .

Theorem 2 . 2 .

 22 For any β > 1, T -β is locally eventually onto on (0, 1] \ G(β), T -1 -β (G(β)) ⊂ G(β) and lim k→∞ λ T -k -β (G(β)) = 0 .

  3) where |w| denotes the length of the word |w|. In particular, |ϕ n (1)| and |ϕ n (2)| are odd, |ϕ n (21)| = 2 n+1 , g n is odd if and only if n is odd, and we can write

FIGURE 2 .Proposition 4 . 2 .

 242 FIGURE 2. The (-β)-transformation for β = 5/4, with set of gaps G 1,0 ∪ G 2,0 and support of the invariant measure F 2,2 ∪ F 2,1 ∪ F 2,0 . Here, t k = T k -β (1). Now, ϕ n (1) ends with 1, which gives T (F n,gn ) = F n,0 \ {1}, i.e., iv. holds. Proposition 4.2. Let 1 < β < γ n , n ≥ 0. Then the elements of F n and G n are intervals of positive length which form a partition of (0, 1]. If n ≥ 1, then

FIGURE 3 .

 3 FIGURE 3. The (-β)-transformation for β = 9/8 with gaps G m,k , support of the invariant measure F 3,0 ∪ • • • ∪ F 3,5 , andt k = T k -β([START_REF] Allouche | A relative of the Thue-Morse sequence[END_REF]. Ratios between G 1,0 , G 2,0 and the rest of the picture are not respected. In reality, G 1,0 is almost 10 times the size of G 2,0 , and G 2,0 is almost 5 times the size of F 2,0 .

Corollary 4 . 3 .Proposition 4 . 4 .

 4344 Let 1 < β < γ n , n ≥ 0. Then T -β I∈Fn(β) I = I∈Fn(β) I and T -1 -β I∈Gn(β) I ⊆ I∈Gn(β) I . If γ n+1 ≤ β < γ n , then F (β) = I∈Fn(β) I (and G(β) = I∈Gn(β) I by definition). Let β > 1 and µ be an invariant measure of T -β which is absolutely continuous with respect to the Lebesgue measure. Then µ(G(β)) = 0.

Proof. Let 1

 1 < β < γ 1 and n as in the statement of the lemma. By Corollary 4.3, we haveF (β) = I∈Fn(β) I if β < γ n , and F (β) = I∈Fn-1(β) I if β ≥ γ n . In the latter case, f ϕ n (1) ≤ f ϕ n-1 (1) implies that F n-1,0 = F n,|ϕ n-1 (1)| ∪ F n,0 , and we obtain, similarly to the proof of Proposition 4.2, that I∈Fn-1(β) I = I∈Fn(β) I. Therefore, we always have F (β) = I∈Fn(β) I.Let first n ≥ 1 be odd. For simplicity, we omit points in {T k (1) | k ≥ 0} in the following statements. We have T gn+1 (F n,0 ) = F n+1,0 ∪ F n,1 and T gn-1 (F n+1,0 ) = T |ϕ n (21)|-1 (1), T |ϕ n (1)|-1 (1) .

FProposition 6 . 1 .Proposition 6 . 2 .

 6162 n,k = F (β) .Here, we have only omitted points in T k (1) | 0 ≤ k ≤ g n (2g n -1) .Let now n ≥ 2 be even. Omitting T k (1) | k ≥ 0 , we have T gn (F n,0 ) = F n,gn ∪ F n+1,0 andT gn-1 (F n+1,0 ) = T |ϕ n (1)|-2 (1), T |ϕ n (21)|-2 (1) . Since β ≥ η n+1 , we have T |ϕ n (21)|-2 (1) ≥ 1/β, thus T gn+1 (F n+1,0 ) ⊇ F n,0 , andT 2gn+2 (F n,0 ) ⊇ F n,0 ∪ F n,1 . Let β > 1 with β 4 = β + 1, i.e., β ≈ 1.2207. Then T 10 -β (1) = T 5 -β (1),and (T nβ (1)) n≥0 is aperiodic.Proof. It is easy to check that T 10 -β (1) = T 5 -β (1). On the other hand, let α be an algebraic conjugate of β satisfying |α| > 1, i.e., α ≈ -0.2481 ± 1.034i, and let σ :Q(β) → Q(α) be the field homomorphism defined by σ(β) = α. Since T β (x) ∈ {βx, βx -1}, we have |σ(T β (x))| ≥ |α| |σ(x)| -1 for all x ∈ [0, 1] ∩ Q(β), thus |σ(T β (x))| > |σ(x)| if |σ(x)| > 1/(|α| -1). We have T 35 β (1) = β 35β 34β 26β 13β 4 ,and one can check that |σ(T 35 β (1))| > 1/(|α| -1). This implies that (|σ(T n β (1))|) n≥35 is a strictly increasing sequence, hence (T n β (1)) n≥0 is aperiodic. Let β > 1 with β 7 = β 6 + 1, i.e., β ≈ 1.2254. Then T 7 β (1) = 0, and (T n -β (1)) n≥0 is aperiodic.Proof. We have T 7 β (1) = β 7β 6 -1 = 0. Let α be an algebraic conjugate of β satisfying |α| > 1, i.e., α ≈ 0.7802 ± 0.7053i, and let σ :Q(β) → Q(α) be the field homomorphism defined by σ(β) = α. We have T -β (x) ∈ {-βx + 1, -βx + 2}, thus |σ(T -β (x))| ≥ |α| |σ(x)| -2 for all x ∈ (0, 1] ∩ Q(β). It is more convenient to consider φ(x) with φ as in the Introduction, since φ(T -β (x)) ∈ {-βφ(x), -βφ(x) -1} and |σ(φ(T -β (x)))| ≥ |α| |σ(φ(x))| -1 for all x ∈ (0, 1] ∩ Q(β).Now one can check that |σ(φ(T 53 -β (1)))| > 1/(|α| -1), which implies that (|σ(φ(T n -β (1)))|) n≥53 is a strictly increasing sequence, hence (T n -β (1)) n≥0 is aperiodic.

  which gives (3.2).

Combining (3.1) for n + 1 and (3.2), we obtain (3.3). Finally, (3.4) follows from (3.1) for n + 1 and n + 2.

  ). By (4.2) and Corollary 4.3, L k consists of the length k suffixes of all concatenations 1 i0 (ϕ(1))

  i1

  • • • (ϕ n (1)) in , i 0 , i 1 , . . . , i n ≥ 0.Therefore, the number of elements in L k is bounded by k n , which grows polynomially in k. This yields lim k→∞

ACKNOWLEDGEMENTS

Lingmin Liao was partially supported by NSFC10901124.

As above, we get T (gn-1)(2gn+2) (F n,0 ) ⊇ gn-1 k=0 F n,k . Since T gn+2 (F n,gn-1 ) ⊇ F n,gn ∪ F n,0 and T gn+2 (F n,k ) ⊇ F n,k+1 for 0 ≤ k < g n , we obtain that T gn(2gn+1) (F n,0 ) = T (gn-1)(2gn+2)+gn+2 (F n,0 ) = gn k=0

PROOFS OF THE MAIN RESULTS

Now we are ready to prove Theorems 2.1 and 2.2.

Proof of Theorem 2.2. For 1 < β < γ 1 , Lemma 5.1 and 5.2 show that T -β is locally eventually onto on Theorem 2.5 was proved in Section 3, thus it only remains to prove Theorem 2.6. Recall that a number

-β be the set of numbers in V -β which are not right endpoints of gaps, and

} is a partition of F (β) up to finitely many points, and T -β (J x ) is a union of intervals J y , y ∈ V ′ -β , for each x ∈ V ′ -β . Define a matrix

i.e., m x,y is the number of times that J y is contained in

Since T -β is locally eventually onto on F (β), the matrix M -β is primitive, thus β is the Perron-Frobenius eigenvalue of M -β , which is a Perron number.

We finally show that the set of Parry numbers and the set of Yrrap numbers do not include each other. For the definition of a Parry number, it is convenient to extend the domain of T β to [0, 1] by setting T β (1) = β -⌊β⌋. Then β > 1 is a Parry number if and only if (T n β (1)) n≥0 is eventually periodic. We know that all Pisot numbers are both Parry numbers [START_REF] Bertrand | Développements en base de Pisot et répartition modulo 1[END_REF][START_REF] Schmidt | On periodic expansions of Pisot numbers and Salem numbers[END_REF] and Yrrap numbers [START_REF] Frougny | On negative bases[END_REF]. Therefore, the symmetric difference between the set of Parry numbers and the set of Yrrap numbers can only contain Perron numbers that are not Pisot numbers.